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Conference Board of the Mathematical Sciences REGIONAL CONFERENCE SERIES IN MATHEMA TICS
supported by the
National Science Foundation
Number 65
MINIMAX METHODS IN CRITICAL POINT THEORY
WITH APPLICATIONS TO DIFFERENTIAL EQUATIONS
Paul H. Rabinowitz
Published for the Conference Board of the Mathematical Sciences
by the American Mathematical Society
Providence, Rhode Island
1 I Iv i ,
, ('\
Expository Lectures
from the CBMS Regional Conference
held at the University of Miami
January 9-13,1984
Research supported in part by National Science Foundation Grant DMS-8303355.
1980 Mathematics Subject Oassifications (1985 Revision). Primary 34C25, 35J60,
47H15, 58E05, 58E07, 58F05, 70H05, 70H30.
Library of Congress Cataloging-in-Publication Data
Rabinowitz. Paul H.
Minimax methods in critical point theory with applications to differential equations.
(Regional conference series in mathematics, ISSN 0160-7642; no. 65)
"Expository lectures from the CBMS Regional Conference held at the University of
Miami. January 9-13. 1984"--T.p. verso.
"Supported by the National Science Foundation."
Bibliography: p.
1. Critical point theory (Mathematical analysis)-Congresses. 2. Maxima and minima
Congresses. 3. Differential equations, Elliptic-Congresses. 1. Conference Board of the
Mathematical Sciences. II. Title. III. Series.
QAl.R33 no. 65 510s [515.3'3) 86-7847
[QA614.7J
ISBN 0-8218-0715-3 (alk. paper)
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Contents
Preface
l. An Overview
2. The Mountain Pass Theorem and Some Applications
3. Some Variants of the Mountain Pass Theorem
4. The Saddle Point Theorem
5. Some Generalizations of the Mountain Pass Theorem
6. Applications to Hamiltonian Systems
7. Functionals with Symmetries and Index Theories
8. Multiple Critical Points of Symmetric Functionals: Problems with Constraints
9. Multiple Critical Points of Symmetric Functionals: The Unconstrained Case
10. Perturbations from Symmetry
11. Variational Methods in Bifurcation Theory
Appendix A
Appendix B
References
v
Preface
This monograph is an expanded version of a CBMS series of lectures ered in Miami in January, 1984. As in the lectures, our goal is to provi introduction to minimax methods in critical point theory and their applic to problems in differential equations. The presentation of the abstract mil theory is essentially self-contained. Most of the applications are to semi elliptic partial differential equations and a basic knowledge of linear ellipti, ory is required for this material. An overview is given of the subject mat Chapter 1 and a detailed study is carried out in the chapters that follow.
Many friends have contributed to my study and organization of this J
rial. I thank in particular Antonio Ambrosetti, Abbas Bahri, Vieri Benci, Berestycki, Halm Brezis, Michael Crandall, Edward Fadell, Suffian Hm Jiirgen Moser, and Louis Nirenberg for their inspiration, encouragement advice. The CBMS conference was hosted by the Mathematics Departmc the University of Miami. Further thanks are due to the members of the dE ment, especially to Shair Ahmad and Alan Lazer for their efficient handli the meeting and their kind hospitality.
vii
1. An Overview
A focus of these lectures is the existence of critical points of real valli< tionals. The most familiar example occurs when we have a continuousl: clltiable map g: Rn ~ R. A critical point of 9 is a point ~ at which g' Frechet derivative of g, vanishes. The simplest sort of critical points of f global or local maxima or minima.
g(x)
~-------------------x
The setting in which we will study critical point t.heory is an infnite sional generalization of the above. Let E be a real Banach space. A m I of E to R will be called a functional. To make precise what we mea critical point of I, recall that I is Frechet differentiable at u E E if then a continuous linear map L = L( u): E -> R satisfying: for any c > 0, tht 8 = 8(c,u) > ° such that II(u + v) - I(u) - Lvi::; cllvll for all Ilvll ::; ( mapping L is usually denoted by J'(u). Note that J'(u) E E*, the dual SJ E. A critical point u of I is a point at which I'(u) 0, i.e.
J'(u)<p = ° for all <p E E. The value of I at u is then called a critical value of I.
In applications to differential equations, critical points correspond tc solutions of the equation. Indeed this fact makes critical point theory an j tant existence tool in studying differential equations. As an example cc the linear elliptic boundary value problem
(1.1) - Liu J(x), x E n,
u = 0,
, /
where here and in future examples n denotes a hounded domain in R n whose boundary. an. is a smooth manifold. Suppose f E C(n). A fUIlction u is a rlassical solution of (11) if 11 E C2(0) n C(n). For such a solution, multiplying
1) by;; E (0) yields
(12) 1(\'11 \';; dx ° ()
after an integration hy parts. Let H'c;2(n) denote the closure of cO'(n) with respect to
~' == (j~ 1\'1112 dX) 1/2 .
If 11 E H'c;·2(n) and satisfies (1.2) for all 'P E CO'(O), then 11 is said to be a weak solution of (1.1). By our ahove remarks, any classical solution of (1.1) is a weak solution. Under slightly stronger hypotheses on f (e.g. f Holder continuous) the converse is also true. Choosing E == w~·2(n), set
( 1. 3) 1 (u) == r (~ ~ f u) dx. in It is not difficult to verify that. 1 is Fn?chet differentiable on E and
(l.4) J'(u)'P 1n(\'u'V'P~f'P)dX for 'P E E. Thus u is a critical point of 1 if and only if u is a weak solution of (1.1).
As was noted earlier, when E = Rn the most familiar sorts of critical points obtained are maxima or minima. In these lectures we will be dealing mainly with functionals which may not be bounded from above or below even modulo finite dimensional subspaces or submanifolds. Such "indefinite" functionals may not possess any local maxima or minima other than trivial ones. For example
let n = (0, 1T) C R, E = \Yo_ and
(1.5)
where / == dl dx. It is not difficult to show that 1 is differentiable on E and has u = ° as a local minimum. For any other u E E and 0' E R,
1(au) = l'r (~2Iu/12 ~ ~\4) dx -> ~OO as ex so 1 is not bounded from below. Furthermore for each kEN, sin kx E E, and
1(sin kx) > '2k2 ~ '2 -> 00 - 4 4
as k -+ 00 so 1 is not bounded from above. Thus it is not obvious that 1 possesses any critical points other than the trivial one u == 0. Nevertheless we will see later as an application of the Mountain Pass Theorem that 1 possesses positive critical values and the same thing is true for higher dimensional versions of (1.5).
, I
As a second example of an indefinite function,,!, system of ordinary differential equations
(1.6) : dp
dt -Hq(p, q),
consider the Hamiltc
where H: R 2n ~ R is smooth, and P and q are n-tllplcs. \Vc are interest, periodic solutions of (1.6). Taking the period to be 27r and choosing E to t appropriate space of 2if functions. solutions of (1.6) are critical p of
(1.7) I(p, q) = [" Ip(t) . q(t) - H(p(I.), q(t))] dl ..
(This will be made precise in Chapter 6.) To see the indefinite nature of ( suppose n 1. Taking Pk(t) = sin kt and qdt) = -- cos kt shows I(pk, qk) = bounded term ~ ±oo as k --; ±oo. Thus I is not bounded from above or be Despite this, as we shall see later, minimax methods can be applied tc functional (1.7) to obtain periodic solutions of (1.6).
There are at least two sets of methods that have been developed to critical points of functionals: (i) Morse theory and its generalizations am minimax theory. For material on "classical" .tI·!orsc theory, see e.g. [Mi, Ch2]. Generalized Morse theories and the so-called Conley index can be fe in the CBMS monograph of Conley ICC] (see also ISm]). Our lectures focus on minimax theory. This subject originated in work of Ljusternik Schnirelman ILLS] although it certainly had antecedents (see e.g. IBill.
What are minimax methods? These are methods that characterize a cri value c of a functional I as a minimax over a suitable class of sets S:
( 1.8) c = inf max J( u 1. AES uEA • ,
There is no recipe for choosing S. In any situation the choice must r€ some qualitative change in the topological nature of the level sets of 1, i.e sets I-l(s) for s near c. Thus obtaining and characterizing a critical value in (1.8) is something of an ad hoc process.
The Mountain Pass Theorem is the first minimax result that we will study statement involves a useful technical assumption--the Palais-Smale conditie that occurs repeatedly in critical point theory. Suppose E is a real Bal space. Let C1(E, R) denote the set of functionals that are Frechet differenti and whose Frechet derivatives are continuous on E. For IE Cl (E, R), we s satisfies the Palais-Smale condition (henceforth denoted by (PS)) if any sequ (urn) C E for which I(um) is bounded and I'(uml --; 0 as m ~ 00 posse a convergent subsequence. The (PS) condition is a convenient way to t some "compactness" into the functional I. Indeed observe that. (PS) im that Kc == {u E EII(u) = c and I'(u) = O}, i.e. the set of critical points ha critical value c, is compact for any c E R. We will see many examples lat, when (PS) is satisfied.
';,
A~ OVER\'IEW
Let Br denote the open ball in E of radius r about 0 and let DBr denote its boundary, ~ow the Mountain Pass Theorem can be stated,
THEOREY!. Let E be a real Banach space and IE C1(E.R). Suppose 11 satisfies (PS), 1(0) = 0,
(1d there exist constants p, 0: > 0 such that IlaB, 2: 0:, and (h) there is an e E E \ DBp such that I(e) S; 0, Then I possesses a critical value c 2: 0: which can be characterized as
c = inf I(u), gEl
where r= {gEC([O,l],E)lg(O) =O,g(l) e}.
This result is due to Ambrosetti and Rabinowitz [AR]. On a heuristic level, the theorem says if a pair of points in the graph of I are separated by a mountain range, there must be a mountain pass containing a critical point between them. Although the statement of the theorem does not require it, in applications it is generally the case that I has a local minimum at O.
A second geometrical example of a minimax result is the following Saddle Point Theorem iR4]:
THEOREM. Let E be a real Banach space such that E = V 8 X, where V is finite dimensional. Suppose IE C1 R), satisfies (PS), and
(13) there exists a bounded neighborhood, D, 0/0 in V and a constant 0: such that IlelD S; ct, and
(14) there is a constant j3 > ct such that Ilx 2: (3. Then I has a critical value c 2: (3. Moreover c can be characterized as
c = inf max I ( u ) . SEf uES .
where r = {S = h(J5)lh E C(J5, E) and h = id on DD}.
Here heuristically c is the minimax of lover all surfaces modelled on D and which share the same boundary. Unlike the Mountain Pass Theorem, in applications of the Saddle Point Theorem generally no critical points of I are known initially. \'ote that (13) and (J,J) are satisfied if I is convex on X, concave on V, and appropriately coercive. Indeed the Saddle Point Theorem was motivated by earlier results of that nature due to Ahmad, Lazer, and Paul [ALP] and Castro and Lazer [CLIo
Both of the above theorems, generalizations, and applications will be treated in Chapters 2--B, In particular in a somewhat more restrictive setting both the ).fountain Pass Theorem and Saddle Point Theorem can be interpreted as special cases of a more general critical point theorem which is proved in Chapter 5.
yl ueh of the remainder of these lectures will be devoted to the study of variational problems in which symmetries playa role. To be more precise, suppose
AC'J OVEH\'lE\V
E is a real Banach space, G is a group of transformations of E into ;;: IE CI(gR), We say I is invariant under G if I(gu) = I(u) for allg EG lL E g As a first example, consider (L5), It is invariant under G == {id where id denotes the identity map on g Note that we can identify (~ Z2, More generally if p(x, 0 is continuous on [0,7r] X R, is odd in
P(x,O Jo(, p(x, t) dt, then
(1.9)
is invariant under G, As another example consider (1. 7), recalling for th that functions in E are 27r periodic, Let 0 E [0, 27r), z (p, q) E E, (ge~t) ) z(t + 0), and G == {gelO E 10,27r)}, Then it is easy to see that r is im under G, YIoreover G can be identified with Sl,
The above examples show that functionals invariant under a group of s;', , tries arise in a natural fashion, It is often the case that such functionals r multiple critical points, Indeed results of this type are among the most faJ ing in minimax theory, The first example of such a theorem goes back t( work of Ljusternik and Schnirelman [LLS], They studied a constrained tional problem, Le, I restricted to a manifold (which must be invariant G) and proved
THEOREM, If IE CI(Rn,R) and is even, then 115n-1 possesses at distinct pairs of critical points,
Subsequently other researchers extended this result to an infinite dimer setting,
Another multiplicity result is provided by the following Z2 symmetric v of the Mountain Pass Theorem [AR, R2]:
THEOREM, Let E be a real Banach space and I E CI(E,R) with I Suppose 1(0) = ° and I satisfies (PS), (1d, and
(12) for all finite dimensional subspace8 E c E, there is an R = R(E that I(u) <; ° for u E E \ BR(El'
Then I possesses an unbounded sequence of critical values,
In order to exploit symmetries of I, one needs a tool to measure the, symmetric sets, i.e. subsets of E invariant under G. Such a tool is provid the notion of an index theory, With the aid of such theories, minimax cl terizations can be given for the critical points obtained in the two theorerr cited, Index theories will be discussed in Chapter 7 and applied to const] and unconstrained variational problems in Chapters 8-9, In particular \I
see that generalized versions of (1.5) satisfy the hypotheses of the symr Mountain Pass Theorem and possess an unbounded sequence of critical p'
The next question we will study is what happens to a functional wh invariant under a group of symmetries when a perturbation is made whi, stroys the symmetry, No general theory has been developed yet to treat
[) A" OVEHVrEW
illiltter, and we will confine om attention to an example from partial differential f'quation:' in Chapter 10.
Our finai topic. cOH'red in ChaptN 1 L concerns ,wplicatiolls of minimax met hods to bifurcation problems. Such problems ,He of interest since bifurcation phcnomcnd uccur in a wide vilriet~· of settings in nature. Consider the map F. R x E -, E. where
( 1.10) F(A. u) = Lv t H(u) - All,
!~ is a rcal Bilnach space, A E R. vEE, L is a continuous linear map of E into E. ilnd H E Cl E) with H(u) = o(lllllll) as u ~ O. ~ote that F(\O) = 0 for illl A E R. We call these zeros of F trivial solutions of F(A.U) O. A point (11.0) E R x E is cillled a hi/meation point for F if every neighborhood of (11,0) contains nontrivial solutions of F()., 11) O. It is well known-see Chapter 11--· that a necessary condition for (11,0) to be a bifurcation point is that 11 E alL), the spectrum of L. Simple counterexamples show this necessary condition is not sufficient. However in Chapter 11, it will be shown that if (1.10) corresponds to an equation of the form 1'(11) = 0, 11 E alL) is also a sufficient condition for (11.0) to be a bifurcation point. Other sharper results will give more information about the nontrivial solutions of F(A, ll) 0 for ()..u) neaf (11,0) both as a function of and as a function of A.
Lastly there are two appendices. The first, Appendix A, is mainly concerned with an important tool called the Deformation Theorem. It is used to help prove all of our abstract critical point theorems. Appendix B contains some technical results which are useful in verifying abstract conditions like IE CI(E, R) or (rS) in a partial differential equations setting.
Some other sources of material on minimax methods and critical point theory in general are [BW, Bg, Ch2, CC, K, LL8, Mi, N2, P2, R2, 82, Va].
2. The Mountain Pass Theorem and Some Applications
In this chapter we will prove the llSUilJ version of the' :'Jollntain Pass Theon""" and give some applications to semilinear elliptic partial differential cquatio,-, The ideas involved in the proof of the Mountain Pass Theorem arc very A key ingredient is the so-called Deformation Theorem, This latter result pIE an important role in all of our abstract minimax results, Since it is ran, ( lengthy and technical in n-ature, we have relegated it to Appendix A (sec Theof(' AA), The following special case is sufficient for the proof of the ylollntain Pa, Theorem,
PROPOSITION 2.1. Let E be a real Banach space, SlLppose I Cl R I and satisfies (PS). For 3. c E R. set Kc 0= {v E[I(lL) =c (' and '= O} OJ
:As '= {1L E ElI(l1) <; s}. If C 15 not a critlcal value of 1. IJwen any E > 0, the exists an c E (0. E) and I] E CnO. 1] x E. E) such that
1° 1](1.11)=11 ifI(l1)rt[c-E,cTE]. 2° 1](1,Ao+o ) C AC- E '
Now we can prove
THEOREM 2.2 (MOUNTAIN PASS THEOREM [AR]) , Let E be a real B( nach space and 1 E 01(E, R) satisfying (PS). Suppose 1(0) = 0 and
(h) there are constants p, a> a sl1ch that IliJBp ~ a, and (h) there is an e E E \ Bp sl1ch that I(e) <; O. Then 1 possesses a critical value c ~ a ' Moreover c can be characterized as
(2.3)
where
c = inf max 1(11). gEr uE9([O.1J)
r = {g E 0([0, 1]. E)/g(O) = 0, g(l) = e}.
PROOF. The definition of c shows that c < 00. If g E r, gUO, 1]) n 8Bp oj 0 Therefore
max 1(11) ~ inf I(w) ~ a l1Eg([O,lj) wEiJB p
via (h). Consequently c ~ 00. Suppose that c is not a critical value of 1. Then il Proposition 2.1 with E == 0'/2 yields E E (0, f) and T/ as in that result. Choose
THE y!Ol'NTAIN PASS THEORE"!
g E r such that
(2.4) max [(u) < c + S uE9([O,1j) -
and consider h(t) == 1')(1. g(t)). Clearly h E C(IO, I], Also g(O) = 0 and 1(0) = 0 < a/2 ::: C E imply h(O) = 0 [0 of Proposition 2.1. Similarly (I( 1) = r: and [(c) ::: 0 imply that h(l) = e. Consequently hEr amI bv (2.3),
c::: [(u).
But by (2.4), g([O, 1]) c AC-l-£ so 2° of Proposition 2.1 implies h([(), 1]) Acc ,
i.e.
(2.6) max [(u) < c - S, uEh([O,lj) -
contrary to (2.5). Thus C is a critical value of [. Although the above proof is very simple, one sees the basic ingredients of a
minimax theorem: (il a family of sets which is chosen to exploit the properties of [ and which is invariant under the deformation map 1')(1, .), (ii) a minimax value c obtained from this family of sets, (iii) topological arguments giving some estimates for c (and showing in particular that c is finite), (iv) an indirect argument based on the Deformation Theorem proving that c is a critical value of [. This framework will be used repeatedly in these lectures. As an almost trivial application of some of these ideas we have
THEOREM 2.7. Let E be a real Banach space and [ E Cl (E, R) satisfying (PS). If [ is bounded from below, then
(2.8) c inf [ E
is a critical value of [.
PROOF. Clearly c is finite. Set S = {{x} Ix E E}, i.e. S is a collection of sets each consisting of one point. Trivially we have
c = inf max[(u). KES uEK
:\ow for any choice ofE, e,g, E = 1, since 1')(1, ,) as given by Proposition 2,1 maps 8 into 8, the argument of Theorem 2.2 shows c is a critical value of [.:;
The remainder of this section consists of several applications of the Mountain Pass Theorem to boundary value problems for semilinear elliptic partial differential equations. Consider
(2.9) - 6u = p(x, u), x EO,
u 0, x E ao, where OcR" is a bounded domain whose boundary is a smooth manifold, In (2.9) and later applications, ~6 could be replaced by a more general second order divergence structure ulliformly elliptic operator. However we prefer to minimize such technicalities.
THE ,,1OC'lTAIK PASS THEORE"j
The fUllction p will always be assumed to satisfy (pd p(x, 0 E c(n x R, R), and (P2) there are constants a I, a2 :::: 0 such that
01 ~ al -ra21~lj, where () ~,,< (n +2)/(n-2) ifn > 2.
If II ~ 1. (P2) can he dropped while if n= 2, it ,mffices that
iP(x. 01 ~ (]1 exp:p(O,
where :p(O~2 ---) () as I~I -, x. The reason for such structural conditi( the following: The functional associated with (2.9) is
10) flu) = r (H'Vu!2 - P(x,u)) in where P(x, 0 = Jo( p(x, t) dt. A natural space in which to treat (2.10) is WJ 2 (O), the closure of Cil(O) with respect to
([ (I'Vu12 + u2) dX) 1/2.
By the Poincare inequality. there is a constant III > 0 such that
III [ u2
dx ~ .~ dx
for all u E E. Hence we can and will take as nom in E.
(2.11) Ilull ([ !'Vuj2 dX) 1/2 .
In order to apply Theorem 2.2 to the functional f given by (2.10), we ha know that fECI (E, R) and critical points of f are weak solutions of (2.£ Appendix B (Proposition B.lO), we prove that when (pd and (P2) are sati f E CI(E, R) and
(2.12) J'(u)'P = [('vu 'V:p - p(x, u):p) dx.
thereby verifying these two properties. For our first application, we further assume (P3) p(x, 0 = o(IW as ~ ---) 0, and (P4) there are constants 11 > 2 and r :::: 0 such that for I ~ 1 :::: r,
() < I1P (X, 0 ~ ~p(x, O· REMARKS 2.13. (i) Hypothesis (P3) implies that (2.9) possesses the 'It
solution" u == O. (ii) Integrating condition (P1) shows that there exist constants (]3,!l4 > 0
that
(2.14)
10 THE \lOUNTAIN PASS THEORE\1
for all x E 0 and ~ E R". Thus since Ii > 2. 0 grows at a "superquadratic" rate and by (P4)' P gro\\'s at a "superlillear" rate as I ~I ~ 00.
(iii) :':ote that if n = 1 and p(x. ~) = e. i.e. if we are in the setting of example (1.5). (pd(Pl) are satisfied.
THEORE\! 2.15. 1/ p satisfies (PI) (Pol), (2.9) possesses a nontrivial weak solutlOn.
PROOF. Let E = WJ·2(fl) and 1 be defined by (2.10). The weak solution I of (2.9) will be obtained as a critical point of J with the aid of Theorem 2.2. Proposition B.lO and (pd-(P2) imply 1 E CI(E,R). Clearly 1(0) = O. Thus we must show 1 satisfies (IJ). (I2), and (PS). To verify (I2), note that by (P4) and (2.14),
(2.16)
for all u E E. where Ifll denotes the measure of fl. Chosing any u E E \ {O}. (2.16) implies
J(tu) = ~ i1
1'VU 12 dx ~ In P(x, tu) dx
:S ~lluI12~tl'a3 r Ilull'dx-+-04Ifll--> ~OO ~ in
(2.17)
as t --> 00. Hence (I2) holds. For (IJ), by (P3), given any E > O. there is a 6> 0 such that I ~I :S 6 implies
IP(x, 01 :s: ~EI~12 for all x E 0. By (P2) there is a constant A = A(6) > 0 such that I~I :::: 5 implies
IP(x, ~)I :S AIEls+ 1
for all x EO. Combining these two estimates, for all ~ E R and x EO,
(2.18) IP(x, 01 :S ~IEI2 -+- AI
Consequently,
(2.19) I) IIul12
via the Poincare and Sobolev inequalities. Choosing Ilu!1 :s: (E/2A)1/(s-l) yields
(2.20) IJ(u)! :S osEIlull 2
Since E was arbitrary, (2.20) shows J(u) o(llul1 2) as u --> O. Therefore
J(u) = !lluil2 ~ J(u) = ~lIu112 -'-o(lluI!2)
as u -; 0 so (Id holds. Next the verification of (PS) here and in later results is simplified with the
aid of the following result whose proof can be found in Appendix B.
PROPOSITIO:\ B 35. Lct.p wt.ls/YiPl) (P2) (wd! hi' is a bounded seqvence m E such thai J'(u m ) -- () ()8 m -
18 precompllet in E.
/)1/ (210
x. then (
By Proposition B.35, to verify (PS), we need only show if(u",)! M J'(u m ) ---+ 0 as m ---+ 00 implies (urn) is a. bounded sequence. For m (2 with 1L Um and T = /clp(x, u)u ~ u) shows (2,21)
'", _ j "" ", _1 ';') (1 1 \ ,. ;' ~ . iVl -j-Il Ii 11 II ::: l\U) ~11 '1111 '1~ \2 ~ I;) jl1J -T-, [)i(J:r
:::(~~) l TelL,!' Tc I"' J{xElll ,lI(xl<>T} , (.rEO iu(xli<r)
By (P4), (2- 1 11- 1) > 0 and the second term on the right in (2.21) is posit The third term is bounded by a constant inrlependently of?n, Hence (2 implies (Urn) is bounded in E,
Lastly note that 1(0) = 0 while for our critical point u, f(ll) ;::: IX > (], He 11 is a nontrivial weak solution of (2.9),
REMARK 2,22, If hypothesis (pd is strengthened to, e,g., (PI) p(x,~) is locally Lipschitz continuous in n X R,
then (PI )'-(P2) imply any weak solution of (2,9) in E is a classical solutiO! (2,9) (see e.g. [Ag]).
COROLLARY 2.23. Under the hypothesis 0/ Theorem 2.15, i/ paisa satis (P'l)' then (2.9) possesses a positive and a ni'9!!tive classical solution.
PROOF. Set p(x, 0 = (] for ~ '::: (] and p(x, 0 = p(x,~) for ~ ;::: O. Let
P(x, ~) = fo~ pix, t) dt.
The arguments of Theorem 2.15 show that
(2.24) 7(u) = ~ IIuI12 ~ 10 P(x, 11) dx
satisfy the hypotheses of the Mountain Pass Theorem. Indeed '15 satisfies (p (P3)' Moreover, (P4) holds for ~ > (] while 0 == P = '15 for ~ '::: O. Hypoth( (P4) was required to help verify (PS) and (h). The above weaker version of ( implies P;::: 11- 1'15 for large I~I and this suffices to get (PS). To also satisfy (j note that (2.14) holds for ~ :2: 0. Thus choosing u E E \ {O} in (2.17) to b, nonnegative function, (12) holds. Consequently, by the Mountain Pass Theor
~ 6.u = p(x, u), x E 0, (2.25)
u = 0, x E 80,
has a weak solution, u 'Ie (]. By (pU and Remark 2.22, 11 is a classical soiut of (2.25). Let A == {x E Oiu(x) < O}. Then by the definition of '15,
!:.u = 0,
u =0,
x E A,
x E 8.11.
12 THE MOUKTAIK PASS THEOREM
Consequently the maximum principle shows 11 0 in A so A =:2). Thus 11 ?: 0 in n. In fact rewriting (2.25) as
- Ll11 - 11= (2.26)
--'---,-) 11. x E n.
11 0, x E an. where aT = max(a,O) and a- = min(a,O), and noting that C1p(x, are continuollR if defined to he 0 at ~ 0, the strong maximum principle [CHI implies that l1(X) > 0 for x E nand dl1(X)/aU < 0 for x E dD. u(x) heing the outward pointing normal to aD.
The negative solution of (2.9) is produced in a similar fashion. REMARK 2.2i. Theorem 2.15 and Corollary 2.23 can be found in [AR]. Earlier
work in this direction was done by Coffman [Col-2] and Hempel [Hel]. RS"lARl':: 2.28. If p = p(O and n > 2, an identity due to Pohozacv [Po] says
(2.29) f (2nP(11) + (2 - n)l1p(l1)) dx = f x· u(x)lvl1I2 dS Jo Jao
for all solutions of (2.9). In (2.29) u(x) is the outward pointing normal to dn. Consequently if n is starshaped with respect to the origin. i.e. x· u(x) ?: 0 for all :r E an, then
(2.30) In P(uJdx?: n 2 In 11p(l1)dx.
By imposing additional conditions, e.g. x· u(x) 'Ie 0 and 11 > () in n so Vl1 # 0 on on. the inequality in (2.30) becomes strict. Taking in particular P'(O =
(8 + 11 then shows .5 < (n + 2)(n - 2)-1 is a necessary condition for there to exist solutions of (2.9). On the other hand, Brezis and Nirenberg IBN] have established some existence results for P( 0 P' (0 + lower order term when" equals the limit exponent (n + 2)(n 2)--1. In work in progress, Bahri imd Coron have shown if n = 3, and P( 0 P' (0 with s the limit exponent and n noncontractible, then (2.9) has a positive solution. If n is an annulus in Rn it is also known that there exists a positive solution of (2.9) for P( 0 = P'to for all values of s. Just what the relationship between the geometry of nand growth conditions on p need be for there to exist solutions of (2.9) remains an interesting open question.
Our next application concerns a nonlinear eigenvalue problem
(2.31) - 2:,11 = ),p(l1) , x En.
11 = O. x E an, where), E R. For a class of such problems, e -e being a good model case, we will show (2.31) has at least two positive solutions for all large ),.
THEOREM 2.32 [ARj. Suppose p satisfies (p~), (P2) and (Po) there is an r > () such that p(~) > () for ~ E (Oor) and p(r) O.
THE :"rOUi'\TAli'\ PASS TIlEOTm:,,1
Then there eXIsts a A > ° such that for all ,\ > ~. (2.31) has at lew claSSIcal solutions wzth are III ll.
PROOF. As with Corollary 2.23, the proof begins by redefining p. Set p p( 0 for ~ E [0, rJ and () otherwise Then 7J satisfies (pi]). (P2), (P:l)
). If 11 is a solution of
(2.33) 6u = AP(U), x E 0,
U = 0, x E all,
the argument of Corollary 2.23 shows {x E ll'iU < O} (:3 find U > 0 j
Similar reasoning proves {x E Olu(x) > r} = 0. Hence 0 < nix) :s; r in U satisfies (2.31). By these observations and Remark 2.22, to find solutio (2.31), it suffices to produce nontrivial critical points of
(2.34) dx
on E = Wd 2(O), £5 being the primitive of p. To study the properties of h" note first that since p satisfies (pd-(p2)'
C 1 (E, R). Moreover (PS) holds for h,: if(um) is a sequence in E with I),(u, M, the boundedness of p implies 1£5(01 :s; a61EI find
(2.35) M2:I),(um)2:~llumI12 j,\la6!olumldx
Applying the Holder and Poincare inequalities, (2.35) shows (urn) is bour in E. Hence (PS) follows from Proposition B,35. ~ote that (2,35) also iill] I A (n) is bounded from below, Hence by Theorem 2.7,
(2.36) bA == inf h,(u) E
is a critical value of h, for all A E R, It may be the case that bA 0 correspon( to the trivial critical point u = 0 of fA' Indeed (P2Hp3) and the arguill' centered around (2,18)-(2.19) show u 0 is a local minimum of fA and satisfies (II!. In fact
(2.37)
for 1,\1 :s; AO and Ilull :s; a7 provided that AO and a7 are sufficiently small. On other hand as in (2.35) the linear growth of £5 in E implies
(2.38)
for all'\ E Rand u E E. Thus for small A (2,37)-(2,38) show h,(u) 2: 0 h,{ Consequently bA = 0,
To obtain a nontrivial solution, let 'P E E \ {O} such that 'Pix) E [0, r) x E O. By (Ps)
,l P('P(x)) dx > 0,
l4 TilE \!()(::\TAI:\ PASS THEOREM
Thus for A sufficiently large, !),(Y) < 0 ,md bA < 0, Define A =: infp > 01 bA < O}. Then for all A > A, the abo\'(' remarks show IA has a criticial value bA < 0 and a corresponding critical point 1!A which is a posith'c solution of (2,33). \loreover since h(1!AJ < 0, h satisfies (h) of Theorem 2,2, Consequently the \lo11Iltain Pass Theorem yields a second critical point rIA of h such that
h(rIA) > 0 > JA(lJ.Al, Clearly rIA is distinct from ]fA and again above remarks is a positive solution of (2.33), The proof is complete,
Our final application in this section is to a problem for which one can establish t.he existence of at least three nontrivial solutions, Consider the equation:
(2,39) ~ll = Aa(x)ll ~ p(X,ll), x E n,
II = 0, x E an. The function a is assumed to be positive and e.g. Lipschitz continuous in O. Associated with (2.39) is the Sturm-Liouville eigenvalue problem:
(2.40) ~ ~1J = /1a(x)1J, x E n,
IJ = 0, x E an. As is well known, (2.40) possesses a sequence of eigenvalllefi (A J ) with 0 < Al < A2 ::; ... ::; AJ ::;. . and AJ ---+ 00 as j ---+ 00, (The number of times an eigenvalue
appears in the sequence equals its multiplicity,) Concerning solutions of (2,39) we have
PROPOSlTIO:" 2,41. Suppose p satisfies (P3) and (pc) (i) () < 0 == p(x, O~-I Jar ~ # 0,
(ii) hE C I (0 x R, R) and ~hdx,~) > 0 Jar ~ # 0, (iii) h(x,O ---+ 00 as I~I ---+ 00 llniJormly Jor x E O.
Then: (P) Jor A ~ AI, (2,39) possesses only the trivial solution u == 0, (2°) For each A> AI, (2,39) possesses a pair oj solutions with ut (resp, the unique solution oj (2.39) with ut > 0 in n (resp. < 0 in D), In fact the maps A ~ are CI for A> Al and u~ ---+ 0 as A ---> AI·
Since p satisfies (P3), u == 0 is a solution of (2,39) for any value of A, The first assertion of Proposition 2.41 follows simply on multiplying (2,39) by u, integrating by parts, and using (pe)(i) , The second statement can be proved in a variety of different ways~see e,g, [RI, St21~and will be omitted here, We will prove the following result which is essentially due to Struwe !St2],
TllEORE!\.{ 2,42. If p satisfies (P3) and (P6), then Jar all A > A2, (2,39) possesses at least three nontrivial solutions,
The proof of Theorem 2.42 will be carried out in a series of steps, First observe that (pe)(iii) implies an a priori bound for solutions of (2,39), Indeed if 11 is a nontrivial solution, either u has a positive (global) maximum or a negative minimum, Assuming the former, if the max is attained at zEn,
o ~ ~611(Z) = Aa(z)u(z) - z)
THE :-'lOUNTAI:\, PASS THEORE:-'l
or
(2.43) Ii( Z)) :S AO( zj.
The same inequality holds at a negative minimum. Hence by (P6)(iii), the a constant M = M(A) > 0 such that IlulIL= :S 1..,1(,\). In fact since A is fi choosing :\ > A there is an !vi depending on :\ such that Ilnll L= :S M for solution of (2.39) with A :s:\. With this observation, p(x, 0 or actually he can be modified for lei :0: M as in Corollary 2.23 or Theorem 2.32. so
hypotheses (p6)(i)-(iii) also hold for the modification 0 c.= 0 a] satisfies
(P2) Ip(x, 01 :S 121 + 122IEI8-1 for some s as in (P2). Moreover solutions of
- 611 A 0.11 p(x, n),
11 = 0, x ao, (2.44)
x O.
for A :S :\ are also solutions of (2.39). It remains to find a third nontrivial solution of (2.44). As usual let 1
Wci· 2 (rl) and define
h.(n) = il [~lv1112 ~an2 - P(x 11)] dx,
where P is the primitive of p. Since (P6)(ii) implies (pd of Proposition B.34 p and (P2) implies (P2), by that result, I), E C2(E.R). Moreover f), satiE (PS) for. suppose
(2.45) f{ :0: h ~ 112 A l dx -'- l dx 2 ill JIl
By (ps)(iii), for any ,8 > 0 there is a 1 = 1(3) :0: 0 such that for all x En i
e E R,
(2.46) P(x, E) :0: l3e 1·
Choosing (3 > AllaIIL=/2, (2.45)-(2.46) show (urn) is bounded in E so (I follows from Proposition B.35.
We want to find a critical point u), of I), other than the known ones 0, given by Proposition 2.4l. Concerning the latter we have:
LEMMA 2.47. If A> A], then of h.
< 0 and
PROOF. The map A --+ is C 1 for A> Aj. Hence
since I~ (un = O. Moreover by Proposition 2.41. 1;,..(0) = O. Hence I),(n~) < 0 for A> A].
are strict lowl mini
--+ 0 as A --+ Al a
16 THE .\lOl'NTAI:\ PASS THEOhEivl
To show that are strict local minima of [A' we will lise a comparison argument. Let 11.' = or Since h E C'2(£, R), for 11 near w,
(2.48) [A(11) = [A (U') + J~(11')(11 - 10') + P;(W)(l1 - W, 11 -tel + 0(1;11
Since J~(w) = 0, to showw is a strict local minimum for h, by (2.48) and Proposition B.34, it suffices to show that
+ (h(]:, w) + dx 2:
for some,) > 0 and all :p E E. The eigenvalue problem
- 6v h(x,tL')v pav, x E \1, (2.50)
v O. x E an. hiL' iL" a solution p = A ilnd l'
"mallcst rigclwalllc to (2.50). U' with v of OIle sign in n. Hence A must be the
?vloreover A can be characterized as [CHI:
(251) -r h(x. w)v2 ) dx
.\ = inf O"'"EE
Comparing (2.51) to
dx
shows thilt ! > A. This fact and (2.52) then show
(2.53) f; (w) :p) 2: ( 1 ~) In
2: (1 ~) 11:p1!2
and the lemma is proved.
With the aid of Lemma 2.4 7, we will find another critical point of [A' Consider
the larger of h(ll~). (If these numbers are equal either will suffice,) Suppose it
occurs at 1l~. Making the change of variables U = 1l-1l\ and setting 4>A(U) =
h(U + ) - JA(l1\), we see that 4>.\ E Cl(£,R), 4>.\(0) = 0, and h satisfies (PS). Moreover Lemma 2.47 and our choice of origin show f.\ satisfies (Jd and
(J 2) since h ) 2: h, (Il ~). Thus the Mountain Pass Theorem shows h has a positiH' critical value. Returning to the original coordinate system, it follows
that h. hiL'i a critical value CA > fA ), where
h(g(t))
and
fA = {g E C(lO, Ii. £)lg(O) = ll~.g(l) = }. ) by Lemma 2.47 and 0 is a critical point of h., a priori
it may be the case that CA = 0 and the corresponding critical point is llA := O. Thus to ensure that 11), is nontrivial it suffices to show that c), < O. To do this,
THE ~10Cj\;TAJ"i PASS THEOH.EM
the fact that A > A2 will be used. \Ve will construct a curve 9A E l\ sue h is strictly negative on 9A([0.1]). Before doing so, some further obsen about lA are needed.
:\Tote first that
(2.54) l),(tu) S t2 1),(u)
for all u E E and t E [0, Ii. Indeed by (P6)(ii)
P(x, == lZ 0 d~ = l p(x, ty)tdy
t2 r h(x,ty)ydYSt 2 rh(x,y)ydy ./0 ./0
(2.55)
for any t E [0, 1J and z E R. Therefore
(2.56) h,(t11)=t2 {~( -A!/(112)d.r+ (P(x,tu)cixSt 2f:,,(I1) ./n ./n
for all 11 E E. :\Text let V denote t.he span of all eigenfunctions of (2.40) correspond
A I and A2 and let fF denote the closure (in E) of the span of the rem
eigenvectors of (2.40). Thus E = V Wand it is easy to check that
( -In cix )1/2
- Aaw2 )
1/2
can be taken as equivalent norms on V and W. Abusing notation somewh will write Ilvll, Ilwll for these norms. Thus for u = v + wEE,
(2.57) h(u)= ~llvI12+~llwI12+ InP(x,v+w)dx.
If h,(u) < 0, (2.57) and (P6)(i) show
(2.58) + 21n pry, v+ w) dx > IIwl1 2
Therefore there exists a 0 = <5 (11) > ° such that
(2.59) Ilvll > (1 + o)ilwll·
Set = v~ + wi- Since h < 0, (2.59) shows II> 0. Now the path 9A E r,\ can be constructed. The path consists of five pa (a+) The first part is the line segment {tutlt E ,Ii}, where 7+livt
and p > ° is free for the moment. It is clear from (2.56) that h < ° 01
segment (b+) Homotopy to 7-'-V;: via T+(1't + 81£'T). (c) Join to T-1';' on aBp n V, where T- II = (I. !\ote that aB
is a connected set since A > A2 implies the dimension of V is at ICilHt 2. (b-) HomotopYT- to as in (b-i-). (a-) Join 7-U), to 11), by a straight line segment. As in (a-i-). here h, <
18 THE \10UNTAIN PASS THEOREM
To complete the proof we must show h < 0 for steps (b=!:) and (c) if p is sufficiently small. For e E [0,1]'
(2.60)
Hence by (2.59) with 01 = mint <'i( uI),o( u~)), and some simple estimates.
(2.61) -I- ' 2 (1 g2 ) (vs: T iJlL'f)) S; p -2 + 2(1 + ,sd 2 + 0(1) S; canst < 0
as p ~ D. Hence fA < 0 for (b±). A similar estimate shows fA < 0 for (c) and the proof of Theorem 2.42 is complete.
RE:vlARK 2.62. For some related applications of the Mountain Pass Theorem see feR, G].
3. Some Variants of the Mountain Pass Theorem
In this brief section we will examine some other versions of the Moun', Pass Theorem as well as some small generalizations. In Theorem 2.2, the crit value c was obtained by minimaxing lover all curves which join ° and e. '] choice of sets is rather arbitrary. There are several other natural classes of with which one can work. E.g. set
and
Define
ro = {g([O, l])lg is 1-1,9(0) = 0,9(1) = e},
r 1 = {K c ElK is compact, connected. and 0, e E K}.
r 2 = {K c EiK is closed, connected. and 0, e E K}.
c, = inf sup I(u), KEr, uEK
i = 0, 1, 2.
Since ro c {g([O,l])lg E r} C r 1 c r2 , it follows that Co 2: C 2: Cl 2: Moreover the proof of Theorem 2.2 shows that Cl is a critical value of I. Not that by Theorem A.4, 1)(1,.) is homeomorphism of E onto E, it follows t 1)(1,.): ri -t r" i = 0,2. With this additional observation the proof of Theor 2.2 also shows Co and C2 are critical values of I.
What is the relationship between these numbers? It is not difficult to see t if K E r 1 and [ > 0, there is a g([O, 1]) E ro such that K lies in a unifc [-neighborhood of g([O, 1]). Thus Co = c = el. We do not know if C2 can di from this common value.
Remaining in the setting of Theorem 2.2 there are other ways in which criti values of I can be characterized that are "dual" in a sense to those that we h, just seen. E.g. let
(3.1) W == {B c E!B is open, ° E B, and e ric B}.
THEOREM 3.2 [AR]. Let I satisfy the hypotheses of Theorem 2.2. Then
(3.3) b == sup inf I(u) BEW uE8B
is a critical value of I with a ::; b ::; c.
19
20 SOME VARIANTS OF THE MOUKTAIN PASS THEOREM
PROOF. By (h), Bp E W. Therefore (3.3) implies b 2: Ct. If g E rand
BE W, there is awE g([O, 1]) aB. Hence
(3A) inf 1::; I(u') ::; max I(9(t)). aB tEIO.I]
Since 9 and B are arbitrary, (3.4) implies that b ::; c. Lastly suppose that b is not a critical value of 1. Then by (the Deformation) Theorem A.4 (with f 0'/2) and Remark A.7(ii), there is an E E (0, f) and ~ E CliO, 11 x E, E) such that
(3.5)
where As = {u E EII(u) 2: s}. Choose BE W such that
(3.6) inf 1> b - f. BB -
By 3° of Theorem AA. ~(l..) is a homeomorphism of E onto E. Therefore
~(1, B) is an open set. Our choice of f and 2° of Theorem AA imply ~(1, 0) = ° and ~(l,e) = e. Hence ° E ~(1,B) and e t,t ~(I,B) = i)(l,EJ). Consequently ~(1. B) E W with ai)(l, B) = i)(1, aB). Thus by (3.5)-(3.6) and the definition
of b.
(3.7) b + f ::; inf 1 ::; b, a(ry(I,B))
a contradiction.
RE)'1ARK 3.8. Simple examples for E = Rn or even R show that b may be
less than c.
REMARK 3.9. The class of sets W of Theorem 3.2 is "dual" to classes intro
duced earlier in that B E W implies that B n K # 0 for all K E foTl' etc. Define
WI = {h(aBp)lh is a homeomorphism of E onto E and h(O) = O,h(e) = e}
and
W2 = {K E ElK n 9([0, 1]) # 0 for each 9 E f}.
Thus W2 :J {aBIB E W} :J WI. Therefore if
b, = sup inf flu), i = 1,2, KEWI uEK
then c 2: b2 2: b 2: bl and it is not difficult to show that bl and b2 are critical
values of I. In fact h = c since for each 9 E L there is a ( = E(g) E E such that
max 1 = I(E). g(IO,I:)
Set K = UgEf E(g)· Then K n g([O, 1]) # (7) for each 9 E r. Therefore K E W2
and infK 1 = c.
SOME VARIANTS OF THE :>.lOUNTAlf\ PASS TBEORE:V!
The next result generalizes Theorem 2.2 by weakening (II!. W refers to (
THEOREM 3.10. Let I E CI(E,R) and satisfy (PS). SlLppose that (Ij) 1(0) = ° and there is (j B E W SlLch that IiaB :::: 0, and (I2l there is an e E E \ E slLch that I(e) SO
hold. Then b defined In (3.3) IS a cntical vallLe of 1 with b :::: 0 [f b = 0, I
exists a critical vallLe of I on CJ B.
PROOF. If b > 0, the proof of Theorem 3.2 with e.g. E = b/2 carries oV( the present situation. Thus suppose b = O. Since 0 E Band e f,i E. without of generality we can assume
(3.11) min(dist(e,E),dist(O,DB)) > 1,
where dist(X, Y) refers to the distance between sets X and Y. Suppose th has no critical value on BB, i.e. l' of 0 on DB. Since the set of critical pc Ko is compact, there exists a neighborhood, 0, of Ko such that 0 tl DB =
By Theorem A.4 with e.g. E = 1 and Remark A .17(iii), there is an E. E (0. 1) r; E C([O, 1] x E, E) such that
(3.12)
In particular by (Ii! and our choice of 0, B B C A_f: \ O. Therefore by (3 and 3° of Theorem A.4, il(I, DB) = Bil(l, B) c Ai: and
(3.13) inf I> E. > O. i)(1,8B) -
We claim il(1, B) E Wand therefore (3.13) contradicts b O. By 3° of Theo: A.4, there is an x E E such that il(I, x) = 0 and by 4° of Theorem A.4,
Ilil(I,x)-xll = Ilxll S 1.
Hence by (3.11), x E Band il(I, B) is a neighborhood of O. Lastly e f,i il(1. Indeed, again by 3° and 4° of Theorem A.4, there is ayE E such that il( = e and Ilil(I, y) - yll = lie - yll s 1 while if y E E, Ily - ell> 1 by (3.11). proof is complete.
REMARK 3.14. Theorem 3.10 was motivated by work of Pucci and Sel [PSI. The following corollary can also be found in [PSI.
COROLLARY 3.15. Let I E CI(E,R) and satisfy (PS). If I has a pail local minima (or maxima), then I possesses a third critical point.
PROOF. Let Ul,U2 be the two critical points and suppose I(lLIl :::: l(v Translating variables so that Ul becomes the origin and replacing I( 1L) by 1* (11 I(u) - I(uIl, we see that l' satisfies the hypotheses of Theorem 3.10 with ( 1L2 - Ul and B the boundary of any small neighborhood of O. Hence I posses a critical value larger than I( uIl or equal to I( lLd in which case I has a criti point on BB.
22 SO:V!E VAR[A~TS OF THE MOUNTAIN PASS THEOREM
RE:V1ARK :U6. If the third critical value equals I( ud, then I has a critical point on aB for all small neighborhoods B of Uj. An argument from point set
topolog~' then shows Kb contains a component of solutions which meets aB for all such small B rWh].
H EMARK 3.17. In roughly the setting of Theorem 2.2 or 3.10, several authors haw studied when any of the corresponding critical points is a "saddle point" of some sort. Taking e.g. E = R" and l(u) = ~ !lul!4 one sees that there need not be any saddle point. Howe\"er several positive results have been obtained. Sec c.g, [Ni. HoZ, PSi.
4. The Saddle Point Theorem
The main goal of this section is to prove the Saddle Point Theorem stated in Chapter 1. The proof requires a simple application of the theory of topological degree in Rn, i.e. Brouwer degree. Since degree theory will also be needed in Chapter 5, we will make a brief digression to discuss this subject. A detailed treatment in the spirit of our exposition can be found in [821.
Let 0 eRn be bounded and open, 'P E CleO, Rn), and bERn \ 'PUJO). The theory of degree is concerned with the existence and mUltiplicity of solutions of the equation
( 4.1) 'P(x)=b.
Suppose 'PI(X) is nonsingular whenever 'P(x) = b. Then the Inverse Function Theorem shows the solutions of (4.1) are isolated. By hypothesis, b t/: 'P(80) and 0 is compact. Hence (4.1) can have only a finite number of solutions in O. For this "nice" case, the (Brouwer) degree of 'P with respect to 0 and b, denoted by drip, 0, b), is defined to be
(4.2) d('P,O,b) = sign det 'PI (x), XE",-l (b)nO
where det A denotes the determinant of a square matrix A. From this definition it is immediate that for such "nice" 'P:
(4.3)
10 d('d 0 b) {I if b EO. ,1, , = 0 if b t/: 0,
2° d( 'P, 0, b) 1= 0 implies there is an x E 0 such that 'P( x) = b, 3° d('P,O,b) =0 ifbt/:'P(O), 4° (Additivity-excision property): If 0 1 , O2 C 0 are open
with 0 1 n 02 = 0 and b 7': 'P(O \ (01 U O2 )), then drip, O,b) = drip, Ol,b) d('P,02,b),
5° (Continuity in 'P): d( 'P, 0, b) = d( 1/),0. b) for all 1}; near 'P Property 5° follows from a straightforward application of the Inverse Function Theorem, In 5°, 1/J near 'P refers to the Cl norm.
The notion of degree given above and properties 1°-5 0 extend to all 'P E C(O, R n) and b ERn \ 'P( 80) with "near" in 5° now referring to the C (0, R n)
2:1
24 THE SADDLE POINT THEOREM
topology. Of course the formula (4.2) is lost in making this extension. A useful and important consequence of 5° is
PROPOSITIOC'{ 4.4. If HE C([O,I] x u,Rn) and b t/: H([CJ,l] x aO). then d(H(t, .), O,b) == constant for t E [0,1].
PROOF. Our above remarks imply that d(H(t, .), O,b) is defined for all t E
[0, I]. By 5° and the definition of degree, the mapping t - d(H(t, -), 0, b) is continuous and integer valued. Since [0, I] is connected. the mapping must. be constant.
COROLLARY 4.5. Ifrp,1;JEC(u,Rn), rp=1iJ onaO, andbERn\rp(30), then d(p, O.b) = d(1/!, O,b).
PROOF. Set H(t, xl = trp(x) + (1 - t)1jJ(x) and invoke Proposition 4.4. The definition of degree given above and its properties extend to an infinite
dimensional setting as follows: E is a real Banach space, ° C E is a bounded and open set, <J> E C(u, E) with <J>(u) u-T(u) and T compact, and b E E\ <J>(30). The resulting degree~still denoted by d( <J>, 0, b )~--is called the Leray-Schauder degree (see [LJ8 or 82]).
With these preliminaries in hand, we turn to the Saddle Point Theorem [R4]:
Tm;OREM 4.6. Let E = V EBX, where E IS a real Banach space and V of {O} and is finite dimensional. Suppose IE CI(E, R), satisfies (PS), and
(I3) there is a constant Q and a bounded neighborhood D of 0 in V such that llaD :S Q, and
(14) there IS a constant (J > Q such that llx 2: .3. Then 1 possesses a critical value c 2: (J . Moreover c can be characterized as
(4.7)
where
c = inf maxI(h(u)). hEr uED
r = {h E C(15,E)lh id on aD}.
PROOF. Let P denote the projector of E onto V obtained from the given splitting of E. If hEr. Ph E C(D, V). Moreover u E 3D implies Ph(u) = Pu = u of O. Identifying V with Rn for some n, d(Ph, D, 0) is defined and by COfo]I'lfY 45 and 10 of (4.3).
d(Ph, D, 0) = d(id, D, 0) = 1.
Hence by 2° of (4.3), there exists an xED such that Ph(x) = O. Consequently for each hEr, there is an x = x(h) E D such that
(4.8) h(x) (id- P)h(x) E X
Hypothesis (14 ) now implies
ma:>Sl(h(u)) 2: I(h(x)) 2: (J. uED
THE SADDLE POINT THEOHEM
Thus by (4,7), c 2: ,3, To show that c is a critical value of 1 requires a fami argument, If Kc = Z, W8 set E = ~(p n:) and invoke Theorem A4 obtainin and 1] as usual. Choose hEr such that
( 4,9) 1(h(u)) <:; (' s,
If 11(1, h) E r, by (4,9) and 7° of Theorem A.4, we get our usual contradicti Since 1](1, h) E C(T5, E), it belongs to r if 1](1, h(v)) v for 11 E 3D, But such 11, h( u) = u and 1] (1, u) 11, This last equality follows via 2° of Theor
A.4 since 1(u) <:; Cl < Cl + € <:; (3 ~ € <:; c + E via our choice of E, The prooj complete,
RE~ARKS 4,10, (i) If V = {O}, (13) makes no sense but (14) and Theor, 2,7 imply 1 has a critical value,
(ii) Hypotheses (13 )-( 14) will be satisfied if e,g, 1 is concave on V, convex X, and appropriately coercive as one approaches infinity along these subspac
Theorem 4,6 was motivated by a PDE existence result due to Ahmad, Laz and Paul [ALP] (see also [Am, AZ]), As an application of Theorem 4,6 a versi of a result in [ALP] will be given next. Consider
(4.11 ) ~ t.u = Aa(x)u + p(x, u), x E (1
u = 0, x E an, where p satisfies (ptl and (P2) with s = 0, If A is not an eigenvalue of (2,3! then (P2) with s = ° and standard elliptic estimates imply an a priori bou for solutions of (4,11) in W5,p(n) for e,g, (3 > n, Then one can use the line elliptic theory for such spaces and the Schauder Fixed Point Theorem to get solution of (4,11), Thus an interesting question to study is what happens wh A is an eigenvalue of (2.40),
THEOREM 4, 12. Suppose A = Ak < Ak-l- I and p satisfies (PI), (P2) Wi s = 0, and
(P7) P(x,O = Jo~ p(x, t) dt -+ 00 as I~I -+ 00 uniformly for x E n. Then (4.11) possesses a weak solution.
PROOF, Let
r (1,~ ,) A " = J \ -2 iVU!- ~2av-
(] \
.\ u)) dx
for u E E WJ,2(O). Since (pJ) and (P2) hold, I E (;1 R), Let V span{vj" where v) is an eigenfunction of (2.39) corresponding to A] ar normalized so that i dx = 1 = A] l dv.
Let X = span{v]iJ 2: k + I} so X = V.L, Therefore E = V X. We will sho that I satisfies (h), (14), and (PS). Then Theorem 4,12 follows from TheoreJ 4.6,
26 THE SADDLE POI;\T THEOREM
(4.13) / in Let ,VI sUPY(Il<E=R Ip(x. ~)I. Then
(4.14) I r P(x.u) dxi <:: M r !ul dx <:: MJllul1 ,in I in
for iilll1 E E Viii the Holder and Poincare inequalities. Combining (4.13)-(4.14) shows 1 is bounded from below on X, i.e. (14) holds. Next if u E V, then u 1I()+1l . where llllE:: E():::Sjxlrl{vJI),j ),dandu EE-=span{vJI),J<),d· Then ( 4.15)
l(u) = ~ I>; (1 - ~k) - J P(x, uo) dx + J (P(x, uO + u" j<k J n 0
Estimating the last term as in (4.14),
(4.16) l(u) <:: -M2 1Iu-11 2 - 10 P(x,uo)dx+MJllu-ll·
:\ow (4.16) and (P7) show l(u) -+ -00 as 11 -+ co in V. Hence I satisfies (h). Lastly to verify (PS), by Proposition B.35 it suffices to show that 11(um )1 <:: K
and 1'(um) -+ 0 implies (urn) is bounded. Writing U m = u~ + u;;; + u;;" where u~ E EO, u;;, E E-, E X, for large m: ( 4.17)±
i!'(um)u;,! = I[ r [vum . vU;' - - p(x, um)u;,1 dxi <:: II. ,}o
Consequently since X = Vl., by (4.17)+, (4.13), (P2 ), and an estimate like ( 4.14),
(4.18) ',i> (1-~) II - )'k+J
which shows that {llu;;'il} is bounded. Similarly (4.17)- shows { II} is bounded. Finally we claim that {lfu~i!} is also bounded. Then (urn) is bounded in E and we are through.
To verify the claim, observe that (4.19)
K::: II(um )! = i10 nUvu;;;1 2 + Ivu;;;12
- ),ka((u;;;)2 + (u;;;)2)]_ (P(x,um ) - P(x,u~))}dx
- 10 P(x, u~) dxl·
THE SADDLE POIlI:T THEOREM
By what has already been shown, the first term on the right is bounded il pendently of m. Therefore
(4.20)
so Un P(x, u~) dx) is bounded. This implies (u~) is bounded. Indeed
LEMMA 4.21. If P satisfies (pd, (P2), and (P7), then
( 4.22) f P(x,v)dx---> (X) 1n as v -t co uniformly for v E EO.
PROOF. By (pd-(p2)' the functional in (4.22) belongs to C I (E, R). By ( for any k > 0, there is a dk such that P(x, 0 ;:: k if I~I ;:: dk for all x E n. v E EO and write v = t'P where 'P E BBl. Then
(4.23) r P(x,t'P)dx;:: r P(x,t'P)dx-Mo, 1n 1nk where Ok nk(t'P)) = {x E OIP(x, t'P(x)) ;:: k} and
Mo ;:: (meas 0) I .inf p(x,OI· xEn,cER
Since 'P =Ie 0, there is an Xo = xO('P) E nand > 0 such that 'P f 0 in B2r (, (the hall of radius 2r about xo). Therefore by (P7), for all large t, 11k :J Br() and P( x, t'P) -t (X) on Br (xo) as t -t 00. Hence the right-hand side of (4.: -t co as t -t 00. Since BBI n EO is compact, (4.22) holds uniformly for v E" and the lemma is proved.
REMARK 4.24. If (P7) is replaced by P(x, 0 -+ -(X) as I~I -+ 00, the abo arguments can easily be modified to handle this case.
5. Some Generalizations of the Mountain Pass Theorem
This section begins with a generalization of the Mountain Pass Theorem in which (Id is weakened. This enables us to treat (2.9) where the nonlinearity also contains a linear part at O. Then an abstract critical point theorem is proved which contains earlier results both of mountain pass and saddle point type. To describe and state it, we will introduce an infinite dimensional notion of linking.
To motivate the first abstract result of this section, consider
- L>1L = '\a(x)lL + p(x.lL),
1L 0, x Eon, J)
x E 0,
where p satisfies (pd-(P4)' This differs from the case treated in Theorem 2.15 due to the presence of the linear term in (5.1). The corresponding functional on E is
(5.2) 1(1L) = 10 [~iVlLI2 ~a1L2 - P(x, 1L)] dx.
If ,\ < '\1, with '\1 as in (2.39), then (fn(lVlL\2 - '\a1L2) dX)I/2 can be taken as a norm on E. 1 satisfies (h), and one can use the Mountain Pass Theorem to establish the existence of a weak solution to (5.1) (and even a positive solution). However if ), > ), I, (11) no longer holds so our previous existence mechanism fails. The next result from [R5] gives us a tool to treat problems where a milder ver8ion of (Id is satisfied.
THEOREM 5.3. Let E be a real Banach space with E = V 9 X, where V IS
finite dimensional. Suppose 1 E CI(E, R), satisfies (PS), and (Ii) there (Ire constants p, Ct > 0 slLch that llaBp r-rX ;:: Ct, (lnd (h) there is (In e E oBI n X (lnd R > p such that if Q == (IJ R n V) EEl {relO <
r < R}, then llaQ s: O. Then 1 possesses a critical value c ;:: Ct which can be characterized as
(5.4)
where
c inf maxl(h(u)), hEr uEQ
r = {h E CCQ. E)lh id on aQ}.
REMARKS 5.5. (il 3Q refers to t.he boundary of Q reiative to V@span{e}.
28
SOME GE:'-1ERALIZATIO);S OF THE ~10U"'TAI]\; PASS THEOREl\l
(ii) If j/ = {O}, then X = E and (I;) reduces to (h). If further 1(0) ) becomes (h). Thus Theorem 5.3 is a generalization of the Mountain]
Theorem. (iii) Suppose Ilv <:; 0 and there is an e E 8B l n X and an R > p such
I( Ii) <:; 0 for u E F ES span{ e} and Ilull 2: R. Then for any large R, Q as deli ill (Is) satisfies IIQ <:; 0.
PROOF OF THEORlcM 5.3. The only novelty in the proof is to show thai (5J)) c 2: o.
Once that has been done, a familiar argument shows that c is a critical v, of I. Indeed if not, set E = 0./2 and invoke Theorem A.4 to get E E (0, E) '1 E C(iO, 11 x E. such that
Choose hEr such that
(5.8) maxI(h(u)\) < c + [. uEQ ., -
Then since I(u) <:; 0 on 8Q, 2° of Theorem A.4 and our choice of E imply t 1)(1, h(u)) = u for u E 8Q. Hence 1)(1, h(u)) E r. But (5.8) and (5.4) then im
c <:; maxI(1)(l, h(u))) <:; C - E, (.LEQ
a contradict.ion. To verify (5.6), it suffices to establish the following interscctwn theorem: PROPOSITION 5.9. If hEr, then
(5.10) h(Q) n 8Bp n Xi- 12.
Given Proposition 5.9, let hEr and w E h(Q) n 8Bp n X. Then
(5.11) maxI(h(u)) 2: I(w) 2: inf I(v) 2: 0, uEQ vEBBpnx
the last inequality following via (Ii). Since h was arbitrary, (5.4) and (5.1 imply (5.6).
PROOF OF PROPOSITION 5.9. Let P denote the projector of E onto V I the given splitting. Then (5.10) is equivalent to
(5.12) {
Ph(u) 0,
II(id P)h(u)11 = p
for some u E Q (depending on h). Expressing u E Q as u = v + re, whE v E B R n V and 0 <:; r <:; R, define
<I>(r, v) = !l1(id - P)h(v + re)ii, Ph(v + re)). Then <I> E C(R x V, R x V). Since hlaQ id, if 1L E 8Q, (5.13) <I>(r,v) = (Ilreli,v) = (r,v),
:lO SO\lE U;;NEInLlZATIONS OF THE \lO,:NTAIN PAS:; THEORF:\l
i.e. <P cc= id on ao In particular <P(r.I·) f (p.O) for u dQ since (Is), (p,O) E O. ld('ntif~'ing R x \' with R" for somc n, wc sec that d(<1J,O,(p,O)) is
defined and h~' 13), Corollary 4;), and F of (401),
(5.14) d(<P,Q, (p,O)) = d(id,Q. (p,O)) = 1.
Hence by 2" of (4.3), there exists u Q such that <p(u) = (p,O), i.e. (5.12) is
satisfied and the proposition is proved.
R E\lARK 5.15. The abo\'(' proof shows r could have been chosen in other
ways. r.g.
{h E C(Q, ;d(h, Q, (p, 0)) f O}.
We now turn to an application of Theorem 5.3 to 1).
THEORE\l 5.16. Suppose p sa.tisfics (pJ! (p,d and (ps) 02'Ojorall~ER.
Then jor all A E R, (5.l) possesses a nontnvial weak solution.
PROOF. If ), < AI, the smallest eigenvalue of (2.40), the result follows from
fCmarks made at the beginning of this chapter. Thus suppose A 2' ),1, e.g.
), E I),b ),k~l]' We will show 1 as defined by (5,2) satisfies the hypotheses of
Theorem 5.3. Clearly 1 E C 1 R) via (p!l(P2). Set V == span{vl,,,.,vd and X = F -"-, where the v) 's arc as following (2.40). For u X,
I) (Ivul 2 ),u2) dx 2' (1 ), IIul1 2
as in (4.1:3) while hy (P.,) (and the proof of Theorem 2.15),
10 P(x, u) dx = 0(liuI1 2) as (L -. O.
Hence 1 satisfies (1;). To verify (15), by Remark 5,5(iii), it suffices to show (a)
Ilv s: 0 and (b) there is an e E oBI n X and R > p such that l(u) s: 0 for
11 E V 8 span{e} == E and Ilull > R. But (a) follows from (ps) and (b) with e.g. e = Vk+l on noting that the argument of (2,16)-(2.17) is uniform for finite
dimensional subspaces of E. It only remains to check that 1 satisfies (PS) or (via Proposition B,35) that
)1 s: },,f and 1'(um) -.0 implies (Urn) is bounded. Choose f3 E (p,-l, 2- 1 ).
Then for m large and 11 = Urn
.'vf +
17)
2' l(u) - 31'(11)U
= r i( ~ - 3)I,vu\2 - ),( ~ - fJ)au2 + fJp(x, u)u - P(x, u)] dx io
2' (~ - 3)llu112 ),( ~ - 3)llaIIL=(o)!luII12(o)
+ (13J.i - 1) 10 P(x, u) dx a5
2' (~-3)lluI12 -),q - J3)llaIiL=(o)!luII12(o) + (3J.i - l)a6Iiulli"(o) - a7,
SOME GEI\ERALIZATIO,\;'; OF THE \1()li'\T.\I\ PASS THEOHE\!
where (P.l) and (2.14) hilH' iWf'1l used. Ih ,[illlCl;lfCj ilH''I'Uliil i"s.sillC(' 11 > ,,> O.
(5.18) U(O) (18 , Ii
lJ : V' Un'
where K(E) ~ x as E ~ O. Choosing" SO Slllilil 111<11 the Ui term ill ( absorbs (with the aid of (5.18)) the L2 terlll. we sec (lIm) is bounded in 1 the proof is complete.
REMARK 5.19. In Corollary 2.2:3, we shows that (2.9) possesses a po solution. The same arguments apply to the current setting if A < A1. HOI
if A 2': AI, we cannot expect a positive solution of (,5.1). Indeed if 1)1 is an ( function corresponding to Al in (2.40), we can assume VI > 0 in rl. Theref 11 is a solution of (5.1),
r (A011 + p(x, 11))111 dx = r (-6u)vI dx in io
= r (-6vdudx = Al r oVludx. io in (5.20)
Consequently
(5.21 ) (AI - A) r 01LVI dx = ! p(x, u)vJ dx. .10 . 0 If u is positive in 0, the left-hand side of (5.21) is nonposit.ive while by (ps) right-hand side is nonnegative. Thus there can only be a positive solution if A = Al and p(X,11(X)) == o.
An examination of our abstract critical point theorems shows that an im tant ingredient in their proofs is an "intersection theorem" which allows L
show that the minimax values defined in these results are indeed critical va E.g. in Theorem 5.3, this is carried out in Proposition 5.9 while in Theor·err it is done in (4.8). Our next goal is to introduce a topological notion of lin which modulo assumptions on the form of the functional will enable us to p a result which contains both Theorems 4.6 and 5.3. It also enables us to we, the splitting assumptions of these theorems.
Thus let E be a real Banach space with E = E1 9 Ez, where both EI E2 may be infinite dimensionaL Let PI, Pz be the projectors of E onto EI associated with the given splitting of E. Set
S == {1> E G([O, 1) x E, E)[1>(O, u) = u and P21>(t, u) =
P2u - K(t, 11), where K: [0,1] x E ~ E2 is compact
Recall K compact means it is continuous and maps bounded sets to relati· compact sets. Let S, Q c E with Q c E, a given subspace of E. Then BQ refer to the boundary of Q in that subspace. We say Sand BQ link if when! 1> E Sand 1>(t,BQ) n S = 0 for all t E [0,1]' then 1>(t, Q) n S of 0 for t E [0, 1). This notion and the examples below are due to Benci and the aut IBR]. For heuristic purposes. one can think of the sets Sand BQ as linkin
32 SOME GE:"ERALIZATlO:"S OF THE :vIOG:"TAl:" PASS THEOREM
every manifold modelled on q and ,hiuing the same boundary intersects S, \\'e will give two examples of such linking corresponding to what occurs in Theorems 4,6 and 5,3,
EXAMPLE 5,22, Let q B n E 2 , where B is a neighborhood of 0 in E 2 ,
E = E 2 , q E q, and S = q + E 1 , Suppose <I> E Sand
(5,23) <I>(t,aq)n(q+Ed :2 for all tE [0,11,
\Yo claim <I>(t, q) n S # 2 for all t E [0,1]' i,e, there is a w = w(t) E q such that
P2<I>(t,w(t)) = q, For u E E2 , oct W(t,u) = P2<I>(t,u) = u - K(t,u), By (5,23),
W(t, u) # q for u E aq, Therefore d(W(t, ,), Q, q) is defined for t E [0,11 and by
the infinite dimensional versions of Propositions 4.4 and 4,3, for any t E [0,11,
(,'),24) d(W(t,), Q, q) d(W(O,), q, q) = d(id, q, q) = 1.
Hence there is a 11'( t) E q as desired and Sand aq link, II EMARK ,5.25, Setting q 0, S = X, and q = D, we are in the setting of
the Saddle Point Theorem, EXA:vlPLE 5,26, Let p > 0, S
Q E iO.rJ]} \".E], e E E1 noB1 , 1'1> p, 1'2> 0, = span{e} E2 , Suppose <I> E Sand
(,),27) <I>(t.aQ) S=2 for all tE [0,1],
\,\'c claim for each t E [0,1]' there is a w(t) E Q such that <I>(t, w(t)) E S, i,e,
P2¢(t, (] and ::P1 <I>(t, p, For u E E2 and r E R, set
'11 (t,(r,
By (,5,27), d(W(/,), q, (p, 0)) is well defined and as in (5,24),
d(W(t,(r,u)),q,(p,O)) = 1.
Consequently there is a wit) as claimed and Sand aQ link,
RE".IARK 5,28, Setting E1 = X, E2 = V, and T1 = R = T2, the relationship
between Example 5,26 and Theorem 5,3 becomes clear.
.'\ow we are ready to state a critical point theorem which unifies Theorems 4,6 and 5,3,
THEORE"1 5,29, Let E be a real Hilbert space with E = EJ ttl E2 and E2 = Et Suppose 1 E C1 (E,R), satisfies (PS), and
(Is) l(u) = HLu, 1l) + btu), where Lu = L 1Pj u + L2 P2 U and L,: E, ~ £, is hounded and selfadjoint, i 1. 2,
(h) b' is compact, and (17) there exist5 a subspace E c E and sets SeE, q c E and constants
o > w such that (i) S c El and lis:::: (x,
q is bounded and lidQ <:: w,
(iii) Sand DQ link. Then 1 possesses a critzcai calue c :::: Q,
SOME GENERALIZATIONS OF THE MOUl\TAI" PASS THEOREM
REMARK 5.30. Roughly speaking, c will be obtained as the minimax
over all surfaces modelled on Q. Although the hypotheses of Theorem 5.2~ more restrictive than in our earlier results because of (1.0 ), (h), £'2 == Et, it is easy to check that the PDE applications presented earlier do indeed sa these conditions.
PROOF OF THEOREM 5.29. The proof follows the same lines as our ea
results. An appropriate class of sets will be introduced and c defined as minimax of lover this class. Let
f == {h E C([O, IJ x E, E)lh satisfies (f J)-(f 3)},
where
(ftl h(O,u) = u, (f2) h(t,u) = u for u E 8Q, and
h(t, u) = eB(t,U)Lu+K(t, u), where e E C([O, IJ xE, R) and K is comp; Note that id E f and if g, h E f so is go h with
egoh = eg(t,h(t,u)) +Oh(t,U)
and
Kgoh(t, u) = e8g (t,h(t,u))L K h( t, u) + Kg(t, h(t, u) J. Also by (f3) and (h),
(5.31)
where u, = Piu, i = 1,2. The mappings in f have an important intersecti property.
PROPOSITION 5 32. If hE f, then
(5.33) h(t, Q) n S Ie 0 for each t E [0,1).
PROOF. Equation (5.33) is equivalent to solving
(5.34)
and via (5.31) the latter equation is equivalent to
(5.35) U2 + e- BL2 K 2 (t, u) = 0,
where K2 = P2K. Define
<P(t, u) == Pjh(t, 1).) + U2 + e- BL2 K 2 (t, u)
so by (5.34)-(5.35), (5.33) is equivalent to
(5.36) <P(t, Q) n S Ie 0 for some t E [0,1].
By its definition, <P E S. Thus to prove (5.36), by (17 )(iii), it suffices to shm that
(5.37) <P(t,8Q) n S = 0 for each t E 10,11.
:34 SO\JE CE\ERALlZATIO\S OF THE \lOC\TAI\ PASS THEOREM
But <p(t. 11) = IL if u E rJQ by (f 2) and aQ·~ oS' =c 7 by (i) and (ii) of (/7). Hence (5.37) holds and the proposition is proved.
:\ow we can define
(5.38) c = inf sup I(h(1. u)) hEr uEQ
By hypothesis (Ic), b is weakly continuous (i.e. if (u m ) converges weakly to u, b( 11 m) -- b( 11)) [K] The boundedncss of Q and (10) then imply SUPuEQ I (u) < x. Since id E r. it follows that,~ < x. By Proposition 5.32 and (i) of (h),
(5.39) c ::: Ct.
To complete the proof. we must show that c is a critical value of 1. Let
5' = ~ (0 - ... 'J. By our usual argument, if cis 1101, a critical value of 1, there is an S E (0,5') and II E CliO, 1] x E, E) given by Theorem AA such that 'I( 1. c
Choose h E f such that
sup I(h(l, u)) :<:: c -;- S. uEQ
TheIl our usual argument leads to a contradiction provided that '10 h E f. By all above remark, th s will be the case if 'I E f. By I O of Theorem AA, 'I satisfies (r Il and by 2° of that theorem, (5.39), (h )(ii), and the choice of E, (f2) holds.
Finally by Proposition A.I8. fl also satisfies (f3)' RE\!ARK 5AO. See [Ni, BR, and HoI] for other such results. Au applicatiou of Theorem 5.29 will be given in the next chapter.
6. Applications to Hamiltonian Systems
This chapter contains applications of Theorem 5.29 to problems in which both EI and E2 are infinite dimensional. Such situations arise in studying the existence of periodic solutions of Hamiltonian systems. For p, q E Rn and H E e l (R2n, R), the system of ordinary differential equations
(6.1)
is called a Hamiltonian system. Setting z = (p, q) and J = (~ -;}), where] is the n-dimensional identity matrix, (6.1) can be written more concisely as
(6.2)
Onc of the basic properties of such systems IS that if z( t) is a solution, H(z(t)) := const, i.e. "cnergy" is conserved. Indeed
(6.3)
via (6.1) or (6.2). It will be shown that (6.2) possesses periodic solutions under certain condi
tions on H. Before giving a variational formulation of (6.2), some preliminary material on function spaces and norms is needed.
Let L2(SI,Rm) denote the set of m-tuples of 27f periodic functions which are square integrable. If z E L2(SI, Rm), it has a Fourier expansion z LJEZ ajei)t, where a) E em, a_J = CiJ ' and LjEZ [2 < 00. Set
:= (2:::(1 + JEZ
and let We,2(SI, Rm) := {z E L2(SI, Rm)llIzllw8.2 < oo}.
We will mainly be interested in () = ~, m = 2n, and E := W Ij2,2 (S 1 , R 2n). For e.g. smooth z (p, q) E E where p and q are each n-tuples, set
(6.4) 12"
A(z) p·qdt. o 35
36 APPLICATIO:--JS TO HA.\IILTO:\IA:\ SYSTEI-IS
Then it is easy to check that
I :; const (I:, :Ji )ccZ
Therefore A extends to all of E as a continuous quadratic form. This extension will still be denotd by A.
Let Cl ..... e2n denote the usual bases in R2n and set
EO '= Sp8.n{el, ... e2n},
Et '= span{(sin]t)ek (cos
and
E- '= span{(sin]t)c, + (cos
, (cosJt)ek -+- (sinJt)ek+nl JEN.I:;k:;n}.
Jtlek - (sin
JEN.I:;k:;n}. Then E = EO E- E-. In fact it is not difficult to verify that E+, E-. EO are respectively the subspaces of E on which A is positive definite, negative definite, cmd null. and these spaces are orthogonal with respect to the bilinear form
r2r
I"i 10 (p~'-+-pq)dt
il$sociated \vith A. Here z = q) and I" = (p, E.g. if z E E+ and I" E E-, 1": = 0 and -+- 1") = A(z) -+- .'1(11. It is also easy to check th8.t Eo,E+,
and E- 8.re mutually orthogonal in U(Sl.R2n). These remarks show that if z = zO + z-'- T z- E E.
(6.5)
SNVP, 8.S an equivalent norm on E. Henceforth we use the norm defined in (6.5) as the norm for E. The spaces EO, E-'-, E- are mutually orthogonal with respect to the associated inner product via our above remarks.
One further analytical fact about E is needed.
PROPOSITIO'-: 6.6. For each 3 E [1,(0), E 13 compactly embedded in there is an D's > 0 such that
16.i)
for all z E E.
PROOF. See iFr or R7]. To obtain periodic solutions of (6.2). we are led to study critical points of an
associated functional OIl E. Suppose H E Cl(R2n.R) and there is an s E (1,00) :iueh that
\6.8)
APPLICATIONS TO HAMILTONIAN SYSTEMS
for all z E R 2n. Then by Proposition B .39,
)I(z) == (2rr H(z)dt E CI(E,R) .fo
as does !(z) defined by
(6.9) !(z) A(z) - )I(z). If Z C I (SlR2n) is a critical point of 1 and <; = (p, E E, a computatic shows that
(2Jr I'(z)<; = 0 = .fo
which implies that z satisfies (6.2). More generally a critical point z of 1 in E wi be a weak solution of (6.2). However a simple regularity argument then sho\\ Z E CI(SI R2n) (see the proof of Theorem 6.10 below). Thus we are intereste in critical points of (6.9).
THEOREM G. 10 [R6]. Suppo.se H E CI (R2n, R) and satisfies (HdH?O, (H2) = o(lzj2) as Izi -+ 0, and (H3J there are constants il > 2 and r > 0 such that 0 < ilH(Z) 'S z . Hz(z
for all ? r. Then for (my T > 0, (6.2) possesses a nonconstant T periodic solution. PROOF. Making the change of variables T = 2'rrtT- 1 , (6.2) becomes
(6.11) dz/dT = ).,JHAz), where)., = (27T)-IT and z is 27T periodic in T. Since this has the same form as (6.2) with H replaced by )"H, without loss of gcncmlity w(' can takc T = 27T and work with (6.2).
Basically the proof reduces to verifying that. Theorem 5.29 is applicable here. However there are some technical complications. Since the growth condition (6.8) has not been assumed for H, the corresponding term in (6.9) need not belong to C 1 (E, R). Thus to get a C I functional on E, H wi!! be modified so that it grows like a power of Izl as [z! -+ 00. Let K > 0 and X E COO(R,R) such that x(y) == 1 if y::; K, X(y) == 0 if y ? K + 1, and X'(y) < 0 if y E (K, K + 1), where K is free for now. Set
(6.12)
where R = R(K) satisfies
R?
Then HK E C I(R2n,R), satisfies (HJ)-(H2) and (6.8) with" = 4. ~loreo\'er a straightforward computation shows (H3) holds with II replaced by v = min(
APPLICATIO:-;S TO HA\llLTOI\IAI\ SYSTEMS
for illl z c R 2'1. where (lJ .11.j > 0 and ilre independent of K. set
(G.14)
We will ,;how II{ satisfies the hypotheses of Theorem 5.29. This willieild to a Ilonconstant 27i periodic (weak) solution 21{ (I) of
(6.15)
It will then be shown that ZK is a classical solution of (6.15). Further estimates then prove there is a Ko such that for all K > KO, II ZK II L= ::; Ko. Therefore Jh(ZK) = H(ZK). Hence for such K, ZK satisfies (6.2).
Turning to a study of (6.14), by Proposition B.39. IK c CI(E,R). Choosing E 1 == E~ ami E2 == EO ~ E-. we see that I K satisfies (I5) of Theorem 5.29 with L, clefined by
for Z E E, aIlCI
b(z) = _12K HK(z) dt.
Proposition B.39 implies that b'(z) is compact. Hence (h) holds. The next three lemmas establish (17).
LE\!\1A 6.16. If H satisfies (H2 ), (Ii) (i) holds for h·
PROOF. Dy (H2), for any E > 0, there is a 6 > 0 such that HK(Z) ::; slzl2 if Izi ::; 6. Since HK (z)lz!-4 is uniformly bounded as Izl ~ 00, there is an M = M(s. K) such that HK(z) ::; Mlzl4 for Izi 2: 6. Hence
(6.17) HK(Z)::; +
for all z E R2n. Therefore by (6.17) and Proposition 6.6,
(6.18) fTC HK(z) dt::; [llzI11, + Mllzll1, ::; (m2 + Ma411z112)llz112
Consequently for z E El = E+,
Choose E
h(z) 2: !lzil 2 - (m2 + Ma411zi12)llz112 1 and p so that 3M a4p2 = 1. Then for z E fJBp n E1 ,
h(z) 2: 1p2 a.
Hence IK satisfies (h)(i) with S fJBpnEl' REMARK 6.19. p and a depend on K through M.
LEMMA 6.20. If H satisfies (Hd and (H3J, then h satisfies (I7)(ii).
PROOF. Let e E BBI n El and set
Q == {relO ::; r ::; rl} EB (Be2 n E2),
APPLICATIONS TO HA\llLTO\,IA\, SYSTE'dS 3'
where r1 and r2 are free for the moment. Define E == span{e} E2 so Q c E Let Z = zO + z~ E Br2 Ii E2 . Then
(6.21) Jf{(Z + re) = r2 -
'\Tow by simple inequalities and orthogonalit.ies:
(" Hf{(z + re) dt 2:' a3 ~2" iz + rei" dt - 27fa4
" 0 .10
(6.22) 2:' a5 (12
" Iz + rej2 dt) ,,/2 - a6
a5 (1h(lzOI2 + Izj2 +
2:' a7(lzOI" + rV) - a6
Combining (6.21)·(6.22) shows
(6.23) 19(z + rel ::; r2 .~
Choose r1 so that
(6.24)
for all r ;::: rj. Set
Since
M = max 'P(r). rE[O,rd
1/J(z) ==
)
v/2
dt - at
uniformly as Ilzll -+ 00, in E 2 , 1/J(z) 2:' M if Ilzll is large enough, e.g. Ilzlj ;::: r2 Therefore by (6.23), if Ilzlj ;::: r2,
(6.25) 19(z + re) ::; M -1/J(z) ::; 0.
Combining (6.24) and (6.25) with the fact that 19 ::; ° on E2 via (Hd, we see 19 ::; ° == won DQ and (h )(ii) holds.
REMARK 6.26. Both rj and T2 and therefore Q are independent of K.
LEMMA 6.27. If Sand Q are defined as in Lemmas 6.16 and 6.20, then S and DQ link, i.e. If{ satisfies (h)(iii).
PROOF. Immediate from the definitions of Sand Q and Example 5.26. The above three lemmas show If{ satisfies (h). '\Tow to be able to use Theorem
5.28, it only remains to verify that 19 satisfies (PS). Thus suppose Ilg(Zm)'1 ::; A1 and Ij.;(zm) --> 0 as m --> 00. Then for large m and z = Zm:
M + IIzl! 2:lg(z) -1Ij.;(z)z = J02"[~z. Hf{z(z) - Hf{(z)] dt (6.28)
2: (2~j - l/~j) l" z· Hf{z(z) dt - Mj ;::: M2 11zlli, - M3
10 APPLICATIOi\S TO HAMILTONIAi\ SYSTEMS
via (H3) and the form of HK . In (6.28). both M2 and M3 depend on K. z = zO + z-'- ~ z', (H3) and simple estimates show
(6.29) "\1 +llzll :: (2- 1 v- 1)v 12rr
Hg(z) dt - M4
\",'riting
"ote that .\16 and ;\h are independent of K via (6.13). Inequality (6.29) can also be written as
(6.30)
\ext taking Z = Zm and <; = in
(6.31)
and using the Holder inequality and (6.7) yields
Hence
(6.32)
1 r2rr 21Iz+I\2 = A'(z)z'" ::; lio Hgz
::; "\lg (llzlli, + 1) II.
via (6.28). Similarly choosing <; -z- in (6.31) yields (6 . .32) with z+ replaced by Z-. Combining these inequalities with (6.30) shows
(6.33) Ilzll ::; M12 (1 + Ilz1l3/4 + Ilzll"l) which implies that (zm) is bounded in E. To show that (zm) is precompact in E. note that
Since EO is finite dimensional. (z~) is precompae!. By the argument of Propo· sition B . .35.
(6.34)
where pi: is the orthogonal projector of E onto E'" and T'"' (z) is compact. Hence ((U4) and IK (zm) ~ 0 as m ~ x implies has a convergent subsequence.
Theorem 5.29 is now applicable and from it we conclude that IK possesses a critical value eK :: a = arK) > 0 with a corresponding critical point ZK = (PI{. qK). It remains to show that Zg is a classicalnonconstant solution of (6.2). Since IK )\. = 0 for all \ E K choosing \ = (',0,0) E W1.2(SI,R2n) shows
(6.35) f2rr[PK' D - qg . c; -lhz(zl{)' c;J dt = O. io In particular if \' == 1. (6.35) implies
APPLICATIOt\S TO HA~IILTO:;IA:\ ~\,::;TE:,l::;
Fourier expansion shows that if y E L 2(SI,R2n), [y] = 0, and ~ E R2n, ther there exists a unique x E WI,2(SI,R2n) such that [x] = E and:i; = y. Th(
choice of HK and Proposition 6.6 show that HKz(ZK) E L2(SI,R2n). Conse
quently choosing y = lHKz(ZK) and ~ = [ZK], there is a unique Z = (p,q) E W I.2(SI,R2n) such that !z] = and
(6.36)
Taking the inner product of (6.36) with le;. where e; = E . R2n' and integrating by parts yields
(6.37) 121<[p.,J; - q <P I] dt = O.
Comparing (6.35) and (6.37) which hold for all s' E W1.2(SIR2n) showE Z ZK E W1,2(SI,R2n) and ZK satisfies (6.15) in an a.e. s{'nse. l3ut ZK E
W1.2(SI,R211) easily implies ZK is Holder continuous of order~. Therefore HKz(Zk) E C(SI,R2n) and (6.15) tells us ZK E CI(SI,R2n) and is a classical
solution of (6.15). If Zf{ == const,
h(Zf{) = _121< HK(ZK)dt:s; 0
via (Hd. But h(ZK):::: arK) > 0 so ZK is a nonconstant solution of (6.15). The last step in the proof is to show that there is a Ko > 0 such that for
all K:::: Ko, IlzKIIL= < K. Then HKAzK) = Hz(ZK) and we have the desired solution of (6.2). By Theorem 5.29 and in particular (5.38)
h(ZK) = CK = inf suph(h(l,z)). hEr zEQ
Since h(t, z) == Z E r, (6.38) Cf{ :s; suph(z)
zEQ
with Q as defined in Lemma 6.20. By Remark 6.26, rl and T2 are independent of K. Then for Z = re + zO + z- E Q,
(6.39) rh
112 - io Hf{(z)dt:s; ri via (H1)' Consequently by (6.38)--(6.39)' CK :s; rr independently of K. As in (6.28), this implies
ri :::: h(ZK) - Vk-(ZK )ZK (6.40)
:::: (Tl _v-I) 121r ZK' HKz(ZK) dt - MI.
Thus (6.40) provides a K independent upper bound for J~" Z f{ . H f{ A Z f{) dt. By (H3 ),
(6.41) HK(d :s; v-II' HKz(r;) + lvh
·j2 ,PI'L1( ATJO:\S TO lL\\IILTO:\L\;\ ",.':;TDlS
for all;; E R2n. Choosing \ = ::K. integrating (641) over [0, 27f), and rccalling
that HI\ ) == rDnst SillCecl\ satisfies a Hamiltonian system, (641) yields
27f H K ( Z 1\) -:: /.1- 1 /' h ( Z 1\ . H 1\ z ( Z K )) dt -j- 27f Ah .0
(6.42)
The right-hand side of (6.42) is hounded from aboY(' independently of K via om
above remarks. Thu., (6.42) and (6.13) yield a K independent LX bound for ZK
imel the proof of Theorem 6.10 is complete.
R E\L\ H h C.4:l. J 11 int Theore]]1 6.10, a caveat is ill order. In its
setting. given an~' T > 0, tlwrc l'xins a IlOllconstcmt solution of (6.2) having period T. However T may not he the minimal period of the solution which may
Ill'Tk 1 for :;OIlle k E: N, k > 1. Under further hypotheses on II, Ambrosetti
and Mancini [AM2], Deng [De], and Ekeland and Hofer [EH] have proved there exists a solution having minimal period T for any T > O.
An easy consequence of Theorem 6.10 is
COHOLLAH.Y 6.44. Under the hypotheses of Theorem 6.10, there exist inJimtely many (hsUnd T periodic .solutions of (6.2).
PROOF. Theorem 6.10 provides one such solution, ZI(t). Suppose its minimal
period is Tkjl Apply Theorem 6.10 again with T replaced by T(2ktl- 1 to get
a nonconstant T(2ktl- l periodic solution Z2(t). Certainly Z2 is T periodic and
it is distinct from ZI since its minimal period is less than tkjl. Repeating this process produces a sequence of distinct nonconstant T periodic solutions of (6.2).
RD1AHK 6.45. There is a much stronger version of Theorem 6,10 [R9]: If HE C l (R 2n,R) and satisfies (H3 ), then for any R,T > 0, there exists aT
periodic solution Z of (6.2) with > R (sec also [Bel]). Simple examples
show that T may not be the minimal period of the solution. E.g. suppose n = 1 and H(z) = G(!z:2), where G is smooth and monotone increasing. Then (6,2) hecomes
and == const for a solution. Setting ~ = p + iq, , satisfies
¢ = 2iG'(ki 2)\
,0 exp(2iG' Therefore if T is the minimal period, T =
1. In particular if G' :2 1, T -:: 7f,
We conclude this chapter with an application of a seemingly different nature,
Consider (6,2) where instead of fixing the period, the energy is prescribed, e,g,
H(z) == L
THEORE~1 6.46 [R61. Suppose H E C l (R2n,R) and H-l(l) is the boundary of a compact starshaped neighborhood of 0 with z· Hz # 0 on H-l (1). Then (6.2) has a periodic solution on H-l(l).
This theorem will be obtained with the aid of Theorem 6,10, First a technical
result.
APPLICATIONS TO HAMILTO\iL\\i SYSTI-:\l~
PROPOSrTIOT\ 6.47. Suppose H.H E Cl(R2n.R) wl.th H 1(1)= H- 1(1)
and Hz,Hz lOon H-l(1). If dt) satisfies
(6.48) ~ = JHz(r;)
and 1(0) E H-l(1), then there exists a reparametrization z(t) of Itt) such that z satisfies (6.2) and z(t) E H-l(l). In particular if \ is periodic, so 18
PROOF. Since H- 1(1) is a level set for Hand Hand Hz,Hz lOon this set, these gradients must be proportional on H-1(1). i.e. there is i1 1/ E
C(H-l(l),R \ {O}) such that Hz(z) l/(z)Hz(z) Now dO) E H-1(l) and (6.48) is a Hamiltonian system so ((t) E H-l(1). Set z(t) == \(r(t)) where r(O) = 0 and r is a solution of
(6.49)
Therefore z satisfies z(O) \(0) E H-l(l) and
i = ~~r = 1/(\"(r(t)))JHz(\"(r(t))) = JHz(z(t)),
i.e. (6.2). To justify the assertion about the periodic case, more care must be taken since
the solution of (6.49) with r(O) 0 may not be unique if 1/ is merely continuous. Suppose \" is T periodic and without loss of generality assume 1/ > O. Then 1/ > 0 on d[O, Til and (6.49) implies there is a first positive value I of t snch that r(l) = T. Set 3(t) r(t) for t E [0, II, and for J E N, set 3(t) = JT + r(t - JI) for t E [JI, U + I)I]. Then S E C 1 . Indeed we need only check what happens at t = I.
lirrlS'(t) = lirrH'(t) = lir!.w(dr(t))) tit tTt tTt
= l/(dT)) = v(\(O)) = liJ'!.lI/(\"(r(t I))) tll
= liJ'!.lr'(t - I) = liJ'!.l3'(t). tll ilt
It follows that z(t) = d3(t)) is I periodic. PROOF OF THEOREM 6.46. By Proposition 6.47, it suffices to find an H
such that H- 1 (1) = H-l (1) and for which the existence of a periodic solution of
(6.48) can be established. The geometrical assumption on H-l(l) leads to the construction of such a function. By the starshaped assumption for each z E R2n, there is a unique w(z) E H-l(l) and a(z) > 0 such that z = a(z)w(z). Indeed a(z) = izi Iw(z)I- 1 Note that a is homogeneous of degree one and is C 1 for z I O. Define H(O) = 0 and H(z) = a(z)4 for z 10. Then H E C 1(R2n,R),
H-1(l) = H-l(I), H 2: 0, H(z) = 0(lzI2) as Izl--. 0, and z·Hz(z) = 4H(z) since H is homogeneous of degree 4. Thus H satisfies all of the hypotheses of Theorem 6.10. Hence for e.g. T = 27f, there exists a nonconstant 27f periodic solution \" of (6.48). It need not be the case that H(<;(t)) == 1. However H(,,(t)) == f31 0 since <; 1= O. Set! = f3- 1/
4 Therefore Hhl) == 1. Moreover
(6.50) h~) = ~IJHz(\) =
44 APPLICATlO;';S TO HAMILTOKIAN SYSTEMS
via the homogeneity of Hz. Finally let t = ,2T and z( T) = ,s-( t). Then z is 271,-2 periodic and by (6.50) satisfies
dz ;- (~ ) dt _ r- ( ) dT Hz II dT - oj Hz z .
Thus z is the desired solution and the proof is complete. REMARK 6.51. It is possible to make a more direct study of the existence and
multiplicity of periodic solutions of (6.2) having prescribed energy using a direct variational approach and ideas from Chapters 7-9. See e.g. [AMI, BLMR, EL, VG, WI·
1)
7. Functionals with Symmetries and Index Theories
Probably the most striking applications of minimax methods are results which establish the existence of multiple critical points of functionals which are invariant under a group of symmetries. In this chapter some machinary to treat such questions will be introduced. To be a bit more precise, let E be a real Banach space, 9 a group of mappings of E into E, and I E C1 (E, R). We say I is invariant under 9 if I(gu) = I(u) for all g E 9 and u E E. As a simple example, wppose I is even, i.e. I(u) = I( -u) for all u E E. Then I is invariant under 9 {id, --id} =:: Z2. As was noted in Chapter 1, a second example is provided by (1. 7), a functional associated with periodic solutions of Hamiltonian systems where 9 =:: 51
The earliest multiplicity result for symmetric functionals is due to Ljusternik and Schnirelmann wo proved [LL8]:
THEOREM 7.1. If IE Cl(E,R) and is even, then Iism-l has at least m distinct pairs of critical points.
In order to prove Theorem 7.1 or other such results, we need a tool to measure the "size" of a symmetric set. (By a symmetric set, we mean one which is invariant under the symmetry group.) The so-called Ljusternik-Schnirelmann category [LL8] was introduced for this purpose. A simpler notion, that of genus, is easier to deal with and will be used here. Genus is due to Krasnoselski [K] although we will use an equivalent definition due to Coffman [Col] (see also [CF]).
Let E be a real Banach space and let E denote the family of sets AcE \ {O} such that A is closed in E and symmetric with respect to 0, i.e. x E A implies -x EA. For A E E, define the genus of A to be n (denoted by ,(A) = n) if there is a map 'P E C(A, Rn \ {O}) and n is the smallest integer with this property .
. When there does not exist a finite such n, set ,(A) = 00. Finally set ,(0) = O. EXAMPLE 7.2. Suppose BeE is closed and B n (-B) 0. Let A =
BU(-B). Then ,(A) = 1 since the function 'P(x) 1 for x E Band 'P(x) =-1 for x E -B is odd and lies in C(A,R \ {O}).
REMARK 7.3. If A E E and ,(A) > 1, then A contains infinitely many distinct points, for if A were finite we could write A = B U (-B) with B as in Example 7.2. But then ,(A) = 1.
45
46 IT\'CTI()\'.\L:; \\ITll Sy\!:-IETHIES A\,D I\,DEX THEOHlES
EXA\1PLE ,.j If n ? I iind A is hOlll('oillorphic to • .,'n by an odd map. then
~.(A) > 1. Otlwrwisr therr i, a mapping; E ('IA. R {Of) 'xitil ; odd. Choose
'ln~' E A sllch that Plr) > O. Then -.ri < 0 and hy th(' Interlllcdiate \'alll(, Theorelll. ; lllust \'i\nish ,olTll'\\'hrre on ,1m' path in A joining .r and ".1'.
i\ contradiction.
The main proper! ies of grnus wrll he listed in thl' nC'xt pl'Oposit ion. For A (- (; and ~ > O. let So(A) denote a uniform 6-neighborhoorJ of A. i.e. X,,(A) ~ {r E £1 IT - A <:;,s}.
PHOI'OSITIO\' 7 .. 'J. Let A. B E [. Then
Ie Normal1zn.tion: If:r f O. !({:r} 0 { -.7}) '" 1.
2c .Hopping property: If there PI?"t.' an odd mop f (= C( A. Il). then') (A) :S -.( Il).
:]0 MonotOrt1Clty property: If A c n. !(A) < :(n) . . 12 Sl1badditivity: :(A U B) <:; -((A) ~ :(13). 5° Continuity property: If A 18 comport. -,(A) < x ond there is !!!l > 0 s11ch
that Nh(A) E (; ond (A)) == -,(A).
PROOF. lOis a special case of Example 7.2. To prove 2°5° we assulTle that
:(A).';(B) <x: the remaining ('Clses are trivial. For 2°, suppose -I(B) = n.
Then there exists a function 'P belonging to C(B. R" \ {O}). Consequently 'P 0 J is odd and in C(A.Rn \ {O}). Therefore -/(..1) :S n '" ,(B). Choosing f = id
in 2° !'ields 30. For 4°. suppose IrA) = III and '/(B) '" n. Then there exist ll1appiniis p E CiA. Rrn \. {O}) and L' t= G(B. R" \ {O}). both odd. By the Tietze Extension Theorem. there are mappings'; t= Rm) and 0 E C(E. Rn) such that';I.\ = P and <vIB = v. Replacing';. u by their odd parts, we can assume ,;,~, are odd. Set f {'). Then f t= C(A U B, Rm+n\ {O}) and
is odd. Therefore -I('..t U B) <:; III ~ n '" -,- A:(B). Lastly to get 5°. for
each x t= A, set r(x) == ~ = r(-x) and Tx = (x) u Br(x) Then
: '" 1 by Example 7.2. Certainly A C UxEA Tx and by the compactness of
A. A c I Tx , for some finite set of points Xl,"" Xk· Therefore i(A) < 00
via 40. If '" n. there is a mapping P E C( A R n \ {O}) with 'P odd. Extend
'P to an odd function'; as in 4°. Since A is compact, there is a {; > 0 such that
.; f 0 OIl.Yo Therefore -f(.Y6 (A)) <:; n ,(A). But by 3°, <:; (A)l so we have equality.
RDIARK i.B. For later arguments it is useful to observe that if < x, 2> -1(..1) - :(B). Indeed A C B so the inequality follows from
3°-4° of Proposition 7.5.
:"iext we will caiculate the genus of an important dass of sets.
PROPOSITION 7.7. If AcE. 0 i., a bounded neighborhood of 0 in R k,
and there exists a mapping h E CIA, aO) with h an odd homeomorphism, then ,(A) '" k.
FCc.:CTIONALS WITH SY~njETR.IES Ac.:l) INDEX THEOhII'>17
PROOF. Plainl), :S k. If =} < k. there is a,:; E Ri {O}) with,:; odd. Then 'P 0 h 1 is odd and belongs 1.0 C(ao. RI \ {O}). nut this is contra,'y to the Dorsuk-Glam Theorelll [82] since k > j. Therefore co {~.
The next result illustrates how s:i1ll111etry provides us with a tool for obtaining int.ersection theorems.
PROPOSfTfOC'l 7.8 [el!. Let X /;e (L ol1bsJ!ILU: 01 1'; 01 wrizlII.c7I.mJrl k and A E [ with ,(A) > k. Then A n X # 0.
PROOF. Writing E V 1) X with V a k rlimensional complcll]('nt of X, let P denote the projector of E onto V. If AnX = 0, ]' E C(A. V \ {O}). Moreover P is odd. Hence by 2° of Proposition 7.0, ,(A) :S ,(PAl. The radial projection of P A into a E 1 n V is another continuous odd map. Hence ,( A) :<: ,(a B J n V) k via Proposition 7.7, contrary to hypot.hesis.
As was mentioned earlier, there arc other ways in which to measure the size of symmet.ric sets, not. only in a Z'2 setting but for morc general group actions. Suppose E is a real J3anach space with a group 9 acting on it. Set.
Fix 9 == {u E EI!lU = u for all q 9}·
E.g. for 9 {id, -id}, Fix9 = {O} while for the Hamiltonian (1.7), Fix 9 consists of the set of 2n tuples of const.ant vectors. Let [ ocnotc the family of invariant subsets of E \ {O}, i.e. A E [ if fJX E A for all (I E 9 ano x E A. We say we have an index theory for 9) if there is a mapping i: [ -+ N U {oo} such that for all A, B E [,
1° Normalization: If x ri Fixy, i(UgE9 !11') = I.
2° Mapping property: If I erA, D) and I is cCjuivariant. i.e. 19 =!il for all 9 E 9, thcn irA) :S i(E).
3° Monotonicity property: If A c B,i(A) :S 7(D). 4° 8ubadditlvity: itA U E) :Si(A) -I- i(D). 5° Continuity property: If A is compact and A II Fix 9 = 2, then < 00
and there is as> 0 such that No(A) E [ and i(No(A)) = itA). REMARK 7.9. If A E [and AnFix9 # 2, then irA) = sUPBEE l(B). Indeed
let x E A n Fix::; and define I: A -+ {x} via I (u) = x for all 17 E A. This map is continuous and eCjuivariant. Hence i( A) ::; i ( {x}) ::; i( A) via the mapping and monotonicity properties of i. But A can be replaced by for e.g. r < ~ I!xll and any B E [ lies in such a set.
The genus, ,. provides us with a simple indcx theory where::; {id. As to other index theories, Benc; has int.roduced an 8 1 versioll of genus [Be]. Cohomological index theories can be found e.g. in iY, FRI-2, FRRI and the references cited there. E.g. an especially general such situation is contained in Fadell-Husseini [FRj. One can also find analogues of Propositions 7.7-7.8 for these theories.
In t.he next two chapters we shall see how these tools can be used to obt.ain multiplicity results for symmetric functionals.
8. Multiple Critical Points of Symmetric Functionals: Problems with Constraints
The following two chapters study the existence of multiple critical points of fllnctionals possessing a Z2 symmetry. The same ideas together with an appropriate index theory can be used to treat other kinds of symmetries. SeC' e.g. :AZ, Be2-3, BF, Bg, Brl--2. Pal-2, Sl-2]. Theorem 1.10, the classical ]'('sult of Ljusternik and Schnirelmann ILLS] is for a constmined funct.ional, i.e.
i! functional on a manifold. In this chapter we will prove that result. as well as an infinite climensional generalization and give an application to (2.31). Chapter 9 treats unconstrained functionals. For either type of problem, the main clifficuJty is to find an appropriate class of sets with respect to which one can minimax the functional. We will see how this can be done in the constrained setting first, following Ljusternik and Schnirelmann. For convenience we restate their result:
THEOREM 8.1. Suppose I E C1 (RTI. R) and is even. Then at least n distinct pairs of critical points.
PROOF. For E = Rn and 1 ~ ) ~ n, define
(8.2) "iJ = {A E ciA c 5 n-
1 and ,(A) :-,> j}.
This family of sets possesses the following properties:
1 c I] ¥ 0. 1 ~ ] ~ n.
2° Alonotonicdy property: ~!l :) ~!2 :) ... :) In'
1 possesses
(8.3) 3° Invariance property: Suppose p E C( 5n - 1
, 5 n - 1) and is odd.
Then 'P' -, IJ' i.e. E IJ wheneyer A E 4° Excision property: If A. E ~;J and BE C with I(B) ~ s < ],
then
Indeed ro follows from Proposition 7.7 with D = sn-l. 2° is trivial, 3° is a consequence of the mapping property of Proposition 7.5, and 4° follows from Remark 7.6.
Define
(8.4) c] = inf maxI(u), AE~I) uEA
48
1 ~ J ~ n.
PROBLEMS WITH COl\'STRAINTS
From the monotonicity property of the 'J' it is clear that Cl :s: C2 :S:. .:S: Cn . 1
will show that cJ is a critical value of Ilsn-l. This fact in itself is not sufficient prove Theorem S.l since some of the minimax values may coincide with only c corresponding critical point. The following proposition together with Rem, 7.3 shows the Cj'S are critical values and we get enough corresponding criti points. :\ote that 1 (lL) = 1'(lL) AlL, where A = (1'(11),1/).
PROPOSITION S. 5. If CJ = ... = CJ+P == c, and kc = {,y E sn-l
and = O}, then ,(ke ) 2: p T 1.
PROOF. Suppose that ,(ke ) :s: p, Then by 5° of Proposition 7,5, then a [; > 0 such that ,(Nt(kc )) :s: p, Hence if .N == No(ke ) n sn-l, by 3° Proposition 7.5, ,U-:V) :s: p. Invoking Theorem A.4 and Remark A.17(iv) w 0== intJV and E = 1, there is all c: E (0,1) and TI E C([O, 1] x sn-l,sn-l) W
II(t, u) odd in 11 and satisfying
(S.C))
Chom;c A E
a contradiction.
TI(l. \ c
By 4° 01' (S.3), A \ /./ E' ') imel
E I)' Therefore by (S.C)) and the definition of c,
c:s: I :s: c c,
REMARKS S,7. (i) The minimax values c) can be given another more geon rical characterization, namely
(S.S) CJ = inf{r E Rh(k) 2:]}.
Thus the c7 's are just those numbers at which the sets A.r change genus. Ind
denote the right-hand side of (S.S) by cJ . If r > c]' ,(k) 2:.7 so Ar E 'J an
(S.9) CJ :s: max 1= r. A,
Thus (S.9) shows cJ :s: cJ . If c] < cJ ' let c = ~ Then there is an A E
such that maxA I :s: c. Therefore > 2: j by 3° of (8.3) but c < (' contradiction.
(ii) There are other ways to obtain critical values of lis define
bk = sup min I(u), AE'Ik uEA
1 :s: k:S: n,
Clearly b1 2: b2 2: .. 2: bn and using Theorem A.4 and Remark A.17(iv) s11 the bk's are also critical values of Ilsn-l. Note that Cl = mins"-' I sill! x E sn-l, {x} U { -x} E ,1, :V10reover c 1 bn . To prove this, it suffices to s that '71 = {sn-1}. If not, there is a set A E In such that A # S'1-l Then there is a point y E sn-l \ A. Without loss of generality we can assume y = ( with 6 the origin in R n-1 The projection map P( 11) = (11 1,. ., Un-l, 0) bell to C(A, Rn-1 \ {O}) and is odd. Therefore by the argument of Proposition
50 PEOBLE\lS \\TfH CO';STHAI';TS
~J(A) 'S 11 -- I. a contradiction. Thus Cj = hn · Similarly en = III = maXSn-l f. \Ve do not kno\\' if r J = Ii" -).1 if.J ~ 1. n. However if one used the cohomological index theory of iFRlj instead of genus and defilled corresponding minima:.:: and
maximin values r;. b;, it call be shown that r; = b;, I' Thus ill this sense the cohomological index is a nicer tool to deal with.
There are manv infinite dimensional generalizatiolls of Theorem 8.1. E.g.
THEOHE\l 8,10. Let E be an dImensional Hilbert space and lei f E
Cl(E,R) be even Suppose r > 0, f\iJJJ, satisjies (P8), and fliw, 18 bounded from below. Then f,)B, possesses lnjimtely many distmet pairs of entieal points.
PHOOF. Define the sets J) iL'i in (8.2) for J E N with 8,,--1 replaced by aE T •
These sets still satisfy properties 1 CAe (again with 8,,--1 replaced by anT)' !\ow
define
(8.11 ) CJ = inf sup [(11), AET] l1EA
.JE N,
Since JjiJlJ, is hounded frolllblll()w, 1:1 / - J(:'. :Vlorccwer (PS) illlplies fe· {11 C
anrlf(u) = c and f!~3B,(U) = O} is a compact set for any c E R. With these observatiom3 and Remark A,17(iv), the argument of Proposition 8.5 proves the theorem as earlier.
REl\IAHK 8.12. The requirement that fia1J e satisfies (PS) is too stringent a condition for applications. Consider e.g.
(813) - ~11 Ap(X.11).
11 = O. X
:r E n. an,
with n as usual. Suppose that p satisfies (pd--(p2) and
(ps) ~p(x. 0> 0 if E f 0, and (pg) p(x,~) is odd in E Let E == \Vci 2 (O) and 11 E E, Set
(8,14) flu) == - f P(x, 11) dx. In
Then f E C 1
flaB, we have
R) by Proposition B.10 and f is even, At a critical point 11 of
(8. = 0 = - f p(x,u)'Pdx io pfYll'Y'Pdx in for all y E E. Choosing:p = 11 and using (Ps) shows
p=I'(ll)U=- (p(x,u)udx<O. io Therefore 11 is a weak solution of (8.13) with A = _p-I This suggests using
Theorem 8.10 to get weak solutions of (8.13) on aB1 (or aET ).
To apply Theorem 8,8, observe first that by Proposition B,lO, f is weakly continuous, i.e. Urn ~ u implies f(u m ) ~ f(u), This implies that flaB, is
PH013LE'vl;; WITH CO:-;;-;THAI\lT~
bounded from below for otherwise t here is it sequence (IL", ) C a 13 1 such tha
l(u",) < -nl. But, being bounded, (u.mJ has a subsequence com'erging weakl in E to U B~ Hence ) ~ l(u) = -oc contrary to 1 E el(E, R)
To ycrify (PS), let (11m) be any sequcnce in aB l such that (l(u m)) is bounde· and (11m) -~ 0 as m ~ x, i.e.
(8.16) ) = 1'(11",)- (T'(l1rn)lI m . 4 0
Since is bounded in E and I' is COlnpact (Proposit.ioll 13.10), along sortl subsequence Uut converges weakly to SOIlH; II E [~ a]](1
1'(l1m) ~ I'(11) - (I'(l1)U.)l1 = o.
By the weak continuity of f, 1(11m) -~ 1(11). If 1(11) f 0, by (Ps), 11 f () Henc
1'(11)11 f () and 1'(l1m )l1m f 0 for large rrL. Consequently (8,16) shows
))
possesses it conycrgcnt subseqnence. However if [(v.) 0, II", Ileed not hav
a convergent subsequence, In fact by the above argument any sequence
which is weakly convergent to 0 has f(l1m) ~ 0 and fi~Bl ., 0 but does not necessarily have a convergent subsequence.
What we have actually shown above is that flaB I satisfies (PS)lor for all c 'F where (PS)loc is defined in Remark A,17(i), Moreover (PS)loc at c is all we nee for the proof of the Deformation Theorem, Thus Theorem 8,10 is also valid wit
(PS) replaced by (PS)loc for each c] defined by (8,11), To apply this result t (8.13), by the remarks just made, we need only verify that cJ < 0 for all j. Bu
this is immediate from (p~) and (8,11), Thus we have proved
THEOREM 8,17. 1fp satisfies (pd,(P2),(PS), and (pg), (8,13) possesses sequence of distinct pairs of weak solutions (Ak' on R x aB r . where Ak ' -(I'(l1k)l1k)-l
COROLLARY 8,18, ck=f(uk)->Oask->x,
PROOF, Let VI", . ,Vk be as in (2.40) and set Ek span{ v], .. , ,vd an Et its orthogonal complement, By Proposition 7,8, if A E ;k, A If;; Therefore
Consequently
(8.19)
(8,20)
52 PROBLEMS WITH CO!,;STRAINTS
By the Gagliardo-Nirenberg inequality [NI], or [Fr],
(8.21) (f )a/2(f )(I-a)/2
Ilull~(A) :::; (}8 in Ivul2 in u2
where a E (0,1) is defined by
1
s+
If u E Et-l'
(8.22) i u2 dx :::; Ak-1 i Ivul 2 dx
as in (4.13). Combining (8.20)-(8.22) shows
(8.23) II(u)1 :::; a4(A;(I-a)(s+l) + Ak 1/2)
for u E Et_l (laB1 . As is well known [CHI, Ak ~ 00 as k -4 00. Hence (8.11), (8.19), (8.20), and (8.23) show I(Uk) -40 as k -400,
9. Multiple Critical Points of Symmetric Functionals: The Unconstrained Case
This section contains two abstract results: Clark's Theorem and a symmetric version of the Mountain Pass Theorem as well as some applications of these results.
We begin with the following version of Clark's Theorem.
THEOREM 9.1 [C). Let E be a real Banach space, IE Cl(E,R) with 1 even, bounded from below, and satisfying (PS). Suppose 1(0) = 0, there is a se: K c E such that K is homeomorphic to SJ~l by an odd map, and SUPK 1< 0 Then I possesses at least J distinct pairs of critical points.
PROOF. We argue almost exactly as in Theorem 8.1. Let
and define
(9.2) Ck = inf sup I(u), AE,k uEA
lSkS).
The sets Ik satisfy properties 1°~4° of (8.3) with sn~l replaced by E. HencE CIS C2 S ... :s; cJ . Moreover Cl > -00 since I is bounded from below and cJ < 0 since ,(K) = J via Proposition 7.7 and 11K < O. The result now follow, from:
PROPOSITION 9.3. If Ck = ... = Ck+J C and Kc == {u E EII(u) = C anc J'(u) = O}, then I(Kc)?> p+ 1.
PROOF. The proof follows almost the same lines as that of Proposition 8.3 and will be omitted. (Use must be made of the fact that 1(0) = 0 and c, < 0 fOJ 1 SiS k. Therefore 0 rf- KCi so Kc, E [ and I(Kc,} < 00 via (PS).)
REMARK 9.4. Actually in [C), Clark does not assume that I is bounded frorr below or that there is a K as above but merely that if Ck is defined as in (9.2) then -(X) < Ck < O. The above proof then shows Ck is a critical value of I The form of the special case given in Theorem 9.2 is perhaps more useful fOJ applications.
Two such applications will be given next (see e.g. [AR, He2, and R2]). The) are related to problems treated in Chapter 2, namely (2.31) and (2.39) .. QC(nsidm
53
54 THE (':\(,O:\STH.-'c!:\ED CASE
first
(9.5) - ~11 = A(a(.r)lJ - plIo u)).
11 = 0. .r e!fl.
where \2 is a.s usual and p satisfies (P'1)' (P3). (p~) there i" a (l > 0 such that a(.r)(j - p(x. 6) ::; 0 for all x E \2.
and (P9).
THEORE,-,! 9. G. Suppose p (p'j). (P:3)' (Ps), (pn), and A> Ak, the kth of (2.40). Then (fl5) possesses at lenst k lilstmet of nontrivial
solutwns.
PHOOF. We begin by modifying the problem in a familiar fashion,. For x E \1. set q(I, 0 = a(x)( p(x.O if (E [0. (11. q(x. 0 = q(I. Ed if ( > (l, and let If be odd in (. Consider
(9.7) - ~11 = Aq(X, u), x E \2.
u = o. x E ao. Arguing ,~s in the proof of Corollary 2.23 shows any solution of (9.7) is a solution of (9.5). Hence to prove Theorem 9.G. it suffices to produce at least k distinct pairs of critical points of
I(u) 1n[~I\}uI2 - AQ(x.u)ldx,
where Q is the primitive of q and E = WOI2(O) as usual. Since q satisfies (p~) and is a bounded function, I E C l (E, R) via Proposition B.lO. Likewise I is bounded from below and (PS) holds as in the proof of Theorem 2.32. Clearly 1(0) = 0 and I is even. Thus the proof of Theorem 9.G follows from Theorem 9.1 once the existence of a set K as in that theorem has been established. Let Vl, ...• Vk
denote the eigenfunctions of (2.40) corresponding to AI, . . , Ak normalized so that
II = 1 = A, In v~ dx, 1 ::; i ::; k.
Set
(9.8)
It is clear that K is homeomorphic to sn-1 by an odd map for any r > O. We claim < 0 if r is sufficiently small. Indeed for small T and u E K, 11[( x) i ::; E1 so Q(u) au2/2 P(x,u). Therefore by (P3),
1 k A k r I(u) = 2' L - 2' L }o1J'fdx o(r2)
1=1 i=l
1 k ( A) =-L 1-~ 2 A,
1=1
for small r. Since A > Ab I(u) < 0 and the proof is complete.
THE C"ICONSTHAI"IED CASE
As a second application of Theorem 9.6. we CXiillliIlC agclill the proiJlcIn t real in (2.33) and Theorem 2.32:
c,u = Ap(U), x EO,
u = 0, x E a~. (9.9)
THEOREM 9.10. Suppose p satisfies (pit), (P3), (P5), and (pg). Then for a J E N, there exists a >:J > 0 such that if A > >:J' (9,lD) possesses at leasl distinct pairs of solutions.
PROOF. As in the proof of Theorem 2.32. it suffices to show that the modifi functional
(9.11)
has the appropriate number of critical points. This in turn will follow frc Theorem 9.l. In the proof of Theorem 2.32, it was already established th hE CI(E,R), is bounded from below, satisfies (PS), and IA(O) = O. CleaJ (pg) implies h is even. Thus we need only verify that for any J E N, there a >:) > 0 such that for each A > >:), there is a set K as in Theorem 9,1. Let be as in (9.8). For sufficiently small r, F(u(x)) = P(u(x)) for all u E K and (ps), P(u(x)) > 0 if u(x) f 0, Therefore
inf r P(u(x)) dx = a > O. "EK in
Choose>: = a- 1r2 Then IAIK::; ~r2 + o(r2) < 0 for small r. The proof complete.
Now we turn to a Z2 version of the Mountain Pass Theorem. Such a rest was stated in Chapter l. A more general version will be given here.
THEOREM 9.12. Let E be an infinite dimensional Banach space and, IE CI(E, R) be even, satisfy (PS), and 1(0) = O. If E = V Ell X, where V finite dimensional, and I satisfies
(I;J there are constants p, a > 0 such that IIDBp0x ? a, and
(12) for each finite dimensional subspace E c E, there is an R = R(E) su
that I::; 0 on E \ BR(E)' then I possesses an unbounded sequence of critical values.
REMARK 9.13, All of our previous symmetric results have used roughly tJ same class of sets: ik = {A E [h( A) ? k} (or the corresponding class on a B to construct critical values. These sets do not suffice for the setting of Theore 9.13. To see why, as a model case consider a functional of the form
(9.14) s + 1
J) dx,
56 THE lJ;\CO,-;STRAI"ED CASE
where 1 < s < (n + 2) (n - ). It will be seen later in this chapter that Theorem
9.12 applies to such a functional. Since A E 'k can be chosen to be an arbitrarily large sphere in span{v).. with Vi as in (2.40). (9.14) shows
(9.15)
On the other hand choosing
inf AEik
I(u) = -x.
where] = J(r) is large enough. and using arguments as in Corollary 8.18 we sec
minuEA I(u) :::: ~r2 Hence
sup minI(u) = 00. AE""1k uEA
Thus a new family of sets must be produced t.o prove Theorem 9.12.
PROOF OF THEOREM 9.12. A sequence of families of sets f m will be introduced and a corresponding sequence (em) of critical values of I will be obtained by taking a minimax of I over each I'm. A separate argument then shows
is unbounded. Suppose V' is k dimensional and l' = span{ e) .. .. ,ek}. For m :::: k, inductively
choose em+ 1 rt span{e), .... em} == Em· Set Rm == R(Eml and Dm == BRm ",Em· Let
(9.16) E)ih is odd ar:d h = id on BBRm
;';Ot8 that id E em for all mEN so em of 2. Set
(9.17) Ih E em, m :::: J, Y E E, and fry) :; m - J}.
The following proposition shows that the sets f J satisfy conditions like (8.3).
PHOPOSITIO:O: 9.18. The sets f) possess the following properties: 1" r J of Z for all} E N. 2° (Monotonicity) rJ~l C f J .
3° (Invarwnce) If p E G(E.E) is odd, andp = id on BBRm " Em for all m :::: J, then p: r J ---> r J .
4° If BE rj , Z E E, and ,(Z) :; s < J, then B \ Z E r J - 8 ·
PROOF. Since id E em for all mEN, it follows that r J f/: 0 for all J' E
N. If B = h(Dm\Y) E rJ~l. then m :.:: j + 1 :::: J, hE em, Y E E, and -1(Y) :; m - + 1) :; m - J. Therefore B E f J . !'Iext to prove 3°, suppose
B =, E f J and p is as above. Then po h is odd, belongs to C( Dm, E), and yO h = id on BBR~ n Em. Therefore yO h E Gm and po h(Dm \ Y) p(E) ~ r). Lastly to get 4°, let B = h(Dm \ Y) E r J and Z E E with ,( Z) :; .) < J \Yc claim
(9.19) B\Z
THE UNCONSTRAINED CASE
Assuming (9.19), note that since h is odd and continuous and Z E [, h- 1 (Z) E [ Therefore Y U h- 1 (Z) E [ and by 4° and 2° of Proposition 7.5,
,(Y U h- 1 (Z)) ::: ,(Y) + ,(h-1(Z)) ::: ,(Y) + ,(Z)
:::m-J+s=m-(J-s).
Hence B \ Z E r)-s'
To prove (9.19), suppose b E h(Dm \(YUh-1(Z))). Then b E h(Dm \Y)\Z c B \ Z c B \ Z. Therefore
(9.20) h(Dm \ (Y U h-l(Z))) c B \ Z.
On the other hand if b E B \ Z, then b = h( w) where
wE Dm \ Y \ h-1(Z) c Dm \ (Y U h-l(Z)).
Thus
(9.21) B \ Z c h(Dm \ (Y U h-l(Z))).
Comparing (9.20)-(9.21) yields (9.19) since h is continuous. Now a consequence of minimax values of I can be defined. Set
(9.22) c) = inf maxI(u), BEf, uEB
JE N.
It will soon be seen that if J > k = dim V, Cj is a critical value of I. The following intersection theorem is needed to provide a key estimate.
PROPOSITION 9.23. IJJ>k andBErJ , then
(9.24) BnaBp nX i= 0.
PROOF. Set B = h(Dm \ Y) where m ~ j and ,(Y) ::: m - J. Let 6 = {x E Dmlh(x) E Bp}. Since h is odd, 0 E 6. Let 0 denote the component of 6 containing O. Since Dm is bounded, 0 is a symmetric (with respect to 0) bounded neighborhood of 0 in Em. Therefore by Proposition 7.7, ,taO) m. We claim
(9.25) h(aO) c aBp .
Assuming (9.25) for the moment, set W == {x E Dmlh(x) E aBp}. ThereforE (9.25) implies W :::J ao. Hence by 3° of Proposition 7.5, ,(W) m and by Remark 7.6, ,(W \ Y) ~ m (m - J) J' > k. Thus by 2° of Proposition 7.5, ,(h(W \ Y)) > k. Since co dim X k, h(W \ Y) n X i= (2) by Proposition 7.8. But h(W \ Y) c (B n aBp). Consequently (9.24) holds.
It remains to prove (9.25). Note first that by the choice of Hm,
(9.26)
Since m > k, aBp n X n Em i= 2. Hence by (1;),
(9.27)
,'i8 THE UNCO:\STRAI:".'ED CASE
Comparing (9,26) and (9,27) shows Rm > p, :"ow to verify (9,25), suppose
x E 00 and E Bp ' If x E Dm there is a neighborhood /If of x such that
hiS) c Bp ' But then x (j. 00, Thus x E oDm (with 0 relative to Em), But on
oDm' h = id, Consequently if x E oDm and h(x) E Bp , ilh(x)l! = = Rm < p contrary' to what we just proved, Thus (9,25) must hold,
R E\lARK 9.28. A closer inspection of the above proof shows that
2:) - k,
COHOLLIllY CJ 29 JI»k'('l rx>O
PROOF, If) > k and BE f l , by (9,24) and (1;), maxuEB I(u) 2: 0:, Therefore by (9,22). c) 2: 0:,
The next proposition both shows cJ is a critical value of I for J > k and makes an appropriate multipicity statement about degenerate critical values.
PROPOSITION 9,30. II) > k, and c) = ' . , = cJ+p == c, then ,(Ke) 2: p + 1.
PROOF, Since 1(0) = 0 while c 2: 0 > 0 via Corollary 9.29, 0 (j. Ke. Therefore Kc E [ and by (PS), Kc is compact. If ,(Ke) <::; p, by 5° of Proposition 7.5, there is a b > 0 such that ,(NIi(Ke )) <::; p, Invoking the Deformation Theorem
with 0 = .YE(Ke) and '[ = 0:/2, there is an E E (0, '[) and T) E G([O, 1] x E, E) such that T)(1..) is odd and
(9.31)
Choose BE
(9,32)
such that
max J (u) <::; C + E uEB
By 4° of Proposition 9,18, B \ () E f), The definition of Rm shows J(u) <::; 0 for
u E OBRm n Em for any mEN. Hence 2° of Theorem A.4 and our choice of E'
imply rl(l, ,) id on OBRm nEm for each mEN, Consequentlyr) (1, B \ ()) E r J
by 3° of Proposition 9,18, The definition of c) and (9,31)-(9,32) then imply
a contradiction"
max J <::; C - E, ry(I,B\Oj
The next proposition completes the proof of Theorem 9,12.
PROPOSITIO?\ 9,33. c) ~ 00 as J -+ 00,
PROOF, By 2° of Proposition 9.18 and (9.22), C)+I 2: c). Suppose the
sequence (c)) is bounded. Then c) ~ C < 00 as J -+ 00. If c) = c for all large j, Proposition 9,30 implies ,(Ke) = 00, But by (PS), Ke is compact so ,(Ke) < 00
via 5° of Proposition 7,5, Thus c > c) for all j E N, Set
K == {u E Eh+1 <::; J(u) <::; c and I'(u) O},
By (PS) again, K is compact and 5° of Proposition 7.5 implies ,(K) < 00 and
there is a 0 > 0 such that ,(N8(K)) = ,(K) == q, Let s max(q, k + 1). The
THE U~CONSTRAIl\ED CASE ,59
Deformation Theorem with c = C, E = C ~ cs , and 0 = N5(K) yields an E and 1/ as usual such that
(9.34) 1/(1, Ac_€ \ 0) C Ac-£'
Choose J E N such that c] > C ~ E and B E r such that
(9,35) maxl<C+E. B -
Arguing as in the proof of Proposition 9.30 shows is in f) as is 1/( 1, B \ 0) provided that 1/(1,,) = id on DBR ", r.Em for all m ,7 But J SOon aBn", nErn for all mEN while c ~ E Cs 2: Ck+l 2: a> 0 via Corollary 9.29 Consequently 1/(1, B \ 0) E f J and by (9.34) (9.35) and the choice of c]'
CJ:c:; m~ I :c:; c ~ [ < c], ~(l,B\O)
a contradiction, The proof is complete. REMARKS 9.36. (i) If E is finite dimensional, the result of Theorem 9.12 also
obtains with the conclusion being that I possesses at least dim X critica points, (ii) There are analogues of Theorem 9.12 when V is infinite dimensional and
when we have an SI rather than a Z2 action (see e.g. !FHRj) , Such a result leads to the generalization of Theorem 6,10 mentioned in Remark 6.45.
As an application of Theorem 9.12, consider
c.u = p(x, u), x E {1
u = 0, x E 30. (9,37)
THEOREM 9,38. Suppose p satisfies (pd, (P2), (P4), and (pg). Then (9.37) possesses an unbounded sequence of weak solutions.
PROOF, With E = W6,2(0) as usual and
l(u) = fo[~lvuI2 ~ P(x,u)] dx,
the proof of Theorem 2,15 shows I E Cl(E, R), satisfies (PS), and 1(0) = 0, Moreover the argument in Theorem 2,15 that showed I satisfied (h) equally well yields (I2)' Clearly (pg) implies I is even. Assume for now that I also satisfies (Ii), Then Theorem 9,12 implies that I possesses an unbounded sequence of critical values Ck = I(Uk), where Uk is a weak solution of (9.37). Since I'(Uk)Uk = 0,
(9,39)
and it follows that
(9.40) Ck = fo[~P(X,Uk)Uk ~ P(x,uk)]dx -> 00
and k -> 00. Hence by (9,39)·-(9.40) and (P4), (Uk) must be unbounded in E and in LOO(O).
60 THE C:\CONSTRAINED CASE
To verify (I;). choose V = span {Vj, ... ,vd where the functions v) are as in (2.40) and X = V ~. By (P2). for all u EO E.
(9.41) flu) :::: I, ~ dx as in 111ls~l dx - a6
By (8.21)(8.22), if u EO BEp n X,
(9.42) 1) - as.
Choose p = p(k) so that the coefficient of p2 in (9.42) is ± Therefore
(9.43)
for 11 E BEp X. Since Ak ~x as k ~ x. p(k) ~ x as k ~ 00. Choose k so that i p2 > 2as. Consequently
(9.44)
and (I;) holds. As a final example of the ideas used in the proof of Theorem 9.12, consider
(9.9) again. In Theorem 9.lD, we proved under appropriate hypotheses on P if A > ),k, (9.9) possesses at least k distinct pairs of weak solutions and these functions correspond to negative critical values of (9.11). Let K be as in (9.8), i.e. K = BET n Ek· By our choice of rand ),k, I < 0 on K. Setting Rm = r
and Dm = Er n Em, define Gm for 1 ::; m ::; k as in (9.16) and define r) as in (9.17) with the further proviso that m ::; k. Proposition 9.18 then holds for j EO [1, k] and m ::; k as does Proposition 9.23. Defining Cj via (9.22), observing that 7)(1..J = id on BET n Ek, and noting that Corollary 9.29 holds with k O. the proof of Proposition 9.30 works equally well in this setting. Thus we have proved:
THEORE:,I 9.45 [AR]. Under the hypotheses of Theorem 9.10, for all A> ),k, (9.9) possesses at least 2k distinct pairs of nontrivial solutions, k pairs corresponding to negative and k pairs to positive critical values of (9.11).
10. Perturbations from Symmetry
In the last two chapters several examples have been given of the existence of multiple critical points for functionals invariant under a group of symmetries. A natural question to ask is: What happens when such a functional is subjected to a perturbation which destroys the symmetry? Some special cases of this question have been studied and while progress has been made, there are not yet satisfactory general answers.
This chapters treats a perturbation question in the setting of Theorem 9.38. Thus consider
(10.1) - 611. = p(x, 11.), x EO,
11. = 0, x E ao, where p satisfies (Pl), (P2), (P4), and (pg). By Theorem 9.38 the corresponding functional has an unbounded sequence of critical values and (10.1) has an unbounded sequence of weak solutions. Suppose f E L2(0) and (10.1) is replaced by
(10.2) 611. = p(x, 11.) + f(x), x EO,
11. = 0, x E ao. The corresponding functional is
(10.3) J(11.) fo[~IV11.12-P(x'11.)-f(X)11.]dX and J is not even if f =f= 0. Nevertheless we have:
THEOREM 10.4 [RB]. lfp satisfies (pd, (P2), (P4), and (pg) and f E L2(0), then (10.2) possesses an unbounded seq11.ence of weak solutions provided that s in (P2) is further restricted by
/3= (n+2)-(n-2)s > _/1_
(10.5) _ ( ) ns-1 /1-1'
REMARKS 10.6.
(10.7)
(il Inequality (10.5) is equivalent to
/1n + (/1 - l)(n + 2) s < 1m + (/1 - l)(n - 2)'
It is easily checked that if s satisfies (10.7), then 8 < (n + 2)(n 2)--] Also, observing that s = 1 and /1 = 2 satisfies (10.7) shows that (10.5) is nonVaC\lOllS.
61
62 PERTT:RBATIO:\S FH()\l SY\I\IETRY
Iii) Slightly Ie,s general versions of Theorem 10.4 were proved by Bahri and
Berestycki :BB. and b:.' Struwc lSI. See also Dong and Li :DL]. Their argu
ments differ fWIIl the one giv'cn herf'. which is somewhat in the spirit of the
s:;nHnetric ~lountain Pil.SS Theorem. For the proof we requirE' an estimate on the deviation from symmetry of I of the form
( 10.8)
for 11 E E. Cnfortunately I does not satisty (10.8): however it can be modified in snch a [,t'ihioll that the new functional, J. satisfies (10.8) and critical
values and points of J are critical value'S and points of I. To motivate the modified problem, a priori bounds for critical points of I in
terms of the corresponding critical values will be obtained. ;'\ote first that by
(P4) there are constants a4, a.o > 0 such that
(10.9)
for all ~ E R.
(10.10)
for all ~ E R.
o 2: a51~!!1 ~ a4
Therefore there is a constant a3 > 0 such that
1 ~(~p(x. 0 + (/3) 2: PIx. 0 + 01 2: (151~!!1 11.
PROPOSITJO:\ 1 (). 11. Under the hypotheses 0/ Theorem 10.4, there eXlsts a wnstant A depending on ::/1:1'(0) such that i/u is a critical point 0/ I,
110.12) r (P(x, u) + 0.4) ciJ: :S A(I(v)2 + 1)1/2 j[] REMARK 10.13. If u is a critical point of (10.3), then (10.10) and (10.12)
easily imply a bound for v in E in terms of I (v). PROOF OF PROPOSITION 10.11. Suppose u is a critical point of I. Then by
(10.10) and simple estimates,
(10.14)
I(v) = I(v) ~ ~I'(v)u = l [~UP(X'lL) ~ P(x,u) ~/v] dx
2: G ~ t) l (up(x, u) + (3) dx ~ ~ IlfIIU(Il) IlvIIL2(O) ~ 0.6
2: a71 (P(x, u) + (4) dx ~ aslluIIL"([]) ~ ae
2: °2711(p(x,v)+a4)dx~a9 and (10.12) follows immediately from (10.14).
To introduce the modified problem, let X E C=(R, R) such that X(O 1 for
E:S 1, X(O := 0 for E 2: 2, and x'(El E (~2,0) for E E (1,2). Set
Q(v) := 2A(J(vJ2 + 1)1/2
and
'I/;(u) := X ( Q(v)-l l (P(x, u) + 0.4) dX) .
PERTURBAT10'iS FJW~1 SY\I\IETllY
:';ote that by (10.12), if u is a critical point of 1. the arg1ll11(,llt of \ Ii", ill ro. ~ and therefore 11)(11) = 1. Finally set •
(10.15) -1 [1. 2 ( I J(u)= 0.21vul Px,u)-v(u)fl1 J dx.
Then J(u) = 1(11) if u is a critical point of I. The following result contains the main technical properties of .1 which we need.
PROPOSITION 10.16. Under the hypotheses of Theorem 10.4: 1° JECl(E,R). 2° There exists a constant (jl depending on II f Ii L' (0) such that
(10.17)
!oralluEE. 3° There is a constant Mo > 0 such that if .1(11) ?: Mo and .1'(11) =,0, then
J(u) = I(u) and 1'(u) = o. 40 There is a constant M 1 ?: M 0 such that for any c > M I, .1 satisfies (P S) lac
at c.
PROOF. Hypotheses (Ptl-(p2) imply I E Cl(E,R). Since X is smooth, the same is true for 1/; and therefore J. To prove 2°, note first that if 11 E su pp 10 (the support of 1/;), then
(10.18)
where Ctl depends on II!IIL'(D). Indeed by the Schwarz and Holder inequalities and (10.10),
(10.19) lin !UdXI ::; IlfllL'(o)llullL'(D) ::; Ct21Iulb(0)
::; Ct3 (In (P(x, u) + a4) dX) III'
If further u E supp 1/;,
(10.20) In (P(x, 11) + a4) dx ::; 4A(I(u)2 + 1)1/2::; Ct4(II(u)1 + 1)
so (10.18) follows from (10.19)-(10.20). Now to get (10.17), by (10.15) and (P9),
(10.21) IJ( u) - J( -u)1 ::; (1/J( 11) + 11)( -u)) lin fu dxl·
To estimate the right-hand side of (10.21), by (10.18),
(10.22) 1/J(u) lin fudxl ::; Ctl1/J(u)(II(uWlfL + 1).
By (10.3) and (10.15),
(10.23) II(u)l::; IJ(u)j lin fUdxl·
64 PERTCRBATJO:\S FROy! SYW,lETIW
Therefore
(10.24) ~ 1 . ii/II \)
Using Young's inequality, the f term on the right-hand side can be absorbed
into the left-hand side yielding
(10.25)
Combining (10.25) with a similar estimate for the term gives (10.17). To prove 3°. it suffices to show that if A10 is large and 11 is a critical point of
J with J(u) 2' ,\10' then
(10.26) Q. \ Ifrp. . \.1 (ui \ (X,llj-a4}CLX<L
n
The definition of v then implies 1,)(1:) == 1 for l' near 11. Hence 1/(11.) 0 so J(u) = J(u), F(u) = J'(u), and 3° follows.
We will show that (10.26) holds. l3y the definition of J,
(10.27) u) (0(u) + 1/J'(u)u)fu] dx,
where'
(llJ.LS) , 2 r /' 2, 1 = X (8(UJ)Q(11)- lQ(u). n up(x, u) dx-- (2A) 8(u)J(u)J (u)u]
and
8(u) = Q- i (u) J (P{x.u) T (L4) dx.
Regrouping terms in (1O.27)-(1O.28) yields
dx (10.29)
where
(10.30)(i)
and
( 1O.:30)(ii)
Consider
(10.31 ) J(u) - ~-----::=--C-7CJ'(u)U.
If = 1 and TI{u) = 0 = 12(11). (10.31) reduces to the left-hand side of (10.14) so (10.26) follows from (10.12). Since 0 'S 1/)(11) 'S 1, if TI(u) and T2(U)
PERTCIWATIO'iS FROM S\';-'!i\!ETHY 65
are both small enough, the calculation made in (10, when carried out for (10,31) leads to (10.12) with A replaced bv a constant which if smaller than 2A, But then (10,26) holds,
It therefore suffices to show that T J (11,), T2 (u) ,~lJ as ,\10 ~ x, If u supp I), Tl(U) = lJ = T2(U), Thus we assume 11 E SUppe}, By (1O.3lJ) and (lO,IS),
(10,32) IT1 (u)l::: 4al(ll(u)11/1l + 1)1[(u)I- 1
We need an estimate relating l(u) and J(u) for u E supp Ij), By (10,3) and (10,15),
Thus by (1O.1S),
(10.33)
l(u) ~ J(u) -Ii Judxl,
l(u) -;- a1 I/1L ~ J(u) - 01 ~ 1Y10 /2
for Mo large enough, If flu) ::: 0, (10.3:3) implies
( 10.34) a1 /1/ + 11(u)l/p ~ Mo/2 + 11(11,)1,
where 1/-1 + IL ~< 1 = L But (10,34) is not possible if Mo is large enough, e,g" Mo ~ 2a11/-1, which we can assume to be the case, Therefore l(u) > 0, Hence (10.33) implies I(u) ~ Mo/4 or l(u) ~ (Mo/4ad l1 , In any event, l(u)-> 00 as Mo -+ x, which together with (10,32) shows T1(U) -+ 0 as Mo -+ 00, Analogous estimates yield T2(U) -+ 0 as Mo ~ 00 and 3° holds,
The verification of 4° follows similar lines to 3°, It suffices to show there is an Ml > Mo such that if (1im) C E, M1 ::: J(um) ::: J{, and <~ 0, then (um ) is bounded, For large m and any p > 0,
pllumil + J{ ~ J(uml - pJ'(um)um
=(~ p(1+T1(um )))llum I12
(10.35) +p(1+T2 (u m )) r u"p(x,um)dx- r P(x,um)cix Jll In
+ [p(1)0(u m ) + T1(um )) w(um )] i fUm cix,
For Ml sufficiently large and therefore T1 , T2 small, we can choose p E (p-1, 2- 1 )
and E > 0 such that
(10,36)
Hence by (10.35), (10,36), and (P4),
(lO.37) il + J{ > ~ ,,' - 2
Using the Holder and Young inequalities, (10,37) implies (um ) is bounded in EAs in Proposition B.35,
D- 1 JI(um) = (1 + Tdum))um - Plum),
66 PEETL"EBATIOi\S FRO\! SYMMETRY
where D: E - E' is the duality map, .D is compact. and !TJ(llm)! S ~ (for ;1,[]
large). Hence (PS) holds.
On the basis of 3" of Proposition 10.16 and the argument of Theorem 9.38, to prove Theorem 10.4 it suffices to show that J has an unbounded sequence of critical points, This we shall do via a series of steps. The first is to introduce a
sequence of minimax values of J. Let the functions ) be as in (2.40) (with e,g,
=: 1), let =: span{ t']" , . ,I'}}, and let be the orthogonal complrment
of E) in E, Comparing J to the functional in Theorem 9,38, the 1:' term in J does not affect the \'crification of (J~) for J, Therefore there is all R) > 0 such
that J(u) S 0 ifll E E) \BR), Set D) =: BR) nE) and
G) =' {h C(D1 , E)lh is odd and h = id on DB R , },
Define
(10,38) J(h(u)), j E N,
These minim;tx \'/\lu('s will llot in general be critical v;tlues of J unless f =, 0,
However we will WiC them a.s part of a comparison argument to prove that .J has an unbounded sequence of critical values, First we will obtain lower bounds for
h} .
P EOPOSITiO\, 10 ':l9, There IS a constant 32 > 0 and kEN such that fOT
all k 2' k,
(1040)
where ,3 was defined in (10,5),
PROOF. Let h E Gk and P < Rk, By Proposition 9,23, there exists a
1J.' E h(Dk)·~ cJBp n J' Therefore
(1041) max J(h(u)) 2' J(w) 2' inf. J(u), uEDk uEBBpnEt_l
To obtain a lower bound for the right-hand side of (10.41), we argue along the
same lines as (9.41)-(9.44) of Theorem 9,38 to choose P = Pk satisfying
1 = const
and therefore
(10.42)
for all u E cJBPk n Et-l' Since Ak 2' a6k2/n for large k [CHI, (10.42) and our choice of Pk yield (10.40) for large k,
To get critical values of J from the sequence (b k ), another set of minimax
values must be introduced, Define
PERTURBATIONS FHO'.1 SY:V!METRY I'" "
and Ak = {H E C(Uk,E)IH!D k E fk and H = id for
u E Qk (3BRk~1 d U ' BIlk)" Ek]}' Set
J(H(u)).
Comparing the definition of Ck to (10.38) shows Ck :> bk ·
PnOPOSITIO:\ 10.43. As,mme Ck > bk :> hi i . For Ii E (0, Ck - od, define
and
( 10.44) Ck(O) inf max J(H(u)). HEAdo) UEUk
Then cd 0) is a critical value of J,
PROOF, The definition of .h(o) implies this set is nonempty, Since Ado) c
Ak ,c,{.5) :>q. SUpposeCk(.5) is not a critical vnJucofJ. Self i(Ck--iJk-5)so 1:' > () and invoke the Dcformat ion Thc:or('1ll obtaining and 1; as llslla.l. Choose
H E ,'cd b) such that
(10.45) J(H(u)) S
Consider T}(l,H(,)). Clearly this function belongs to C(lh, Moreover if u E Qk, H(u) = u since H E ih and therefore J(H(u)) SO via the definitions of Rk and Rk+l, Since we can assume bk :> AI] > 0 and ck(5) :> Ck > bk , the choice of f implies TI(1, H(u)) = u on Qk, Therefore r;(l, H()) E Ak, Moreover since H E Ak (5), if u E Dk ,
J(H(u))Sb k +ESck-E'Scd5) E'
by the choice of f. Therefore by 2° of Theorem A.4, T}(l, H()) E AdS), Consequently by (9.45) and T of Theorem A.4
ma?C J(T}(l, H(u))) S q(6) E, UEVk
contrary to (10.44). Now we are nearly through with the proof of Theorem 10.4. If Ck > bk for
a sequence of k's --> 00, by Proposition 10.43 and 10,39, J has an unbounded sequence of critical values and the proof is complete. It remains to show that Ck = bk for all large k is impossible.
PROPOSITION 10,46, If Ck = bk for all k :> k*, there exists a constant 0J > 0 and k ~ k * such that
(lO.47)
for all k ~ k. Comparing (lO.47) to (10.40) and (10,5) yields a contradiction and the proof
of Theorem lOA is complete,
68 PERTURBATIONS FROM SYMMETRY
PROOF OF PROPOSITIO'." 10.46. Let c > 0 and k 2: k'. Choose H E '\.k such that
( 10.48) max J(H(u)) s: bk + f uE[h
Since D k + 1 = Uk U (-Uk), H can be continuously extended to Dk_ 1 as an odd function. Therefore by (10.38),
(10.49) bk + 1 s: max J(H(u)) = J(H(w)) uED k + 1
for some 11' E Dk+ l . If 11' E Uk, by (10.48)-(10.49),
( 10.50) J(H(w)) s: bk -1- E.
~uppose 1)' E -[h. Then since bk -> CXl as k -> (Xl via (10.40), (10.49) and (10.17) imply J(-H(w)) > 0 if k is large, e.g. k 2: k. By (10.17), the oddness of H. and (10.48),
J(H(11')) = J( -H(-w)) s: J(H(-w)) + PI ((J(H( -W)))I/I' -1- 1) ( 10.51)
s: bk + E -1- 01((bk + ElI/1' + 1).
Combining (10.49)-( 10.51) yields
( 10.52) bk _ 1 S:bk €+3d(bk -+- -rl).
Since f is (10.52) implies
( 10.53) bk+! s: bk -i- 31 (b~/1' -1- 1)
for all k 2: k. It remains to show that (10.53) implies (10.47). This will be done by induction.
Suppose (10.47) holds for all k E [k,]] n N. We claim it also holds for j + 1. \Vithout loss of generality, we can assume j 2: 2k and
By (10.53),
(10.54)
bk I ~ > max, +
- o<;l<;k (k + 1)1'11'+1
bJ+ I s: b
J +;31 ((wjl'll'-1 )1/1' + 1)
J
s: bi; + ,31 '2.)w l/ l'll/l'-1 + 1) l=k
J
s: bi: +81 (j - k -r 1) + Plw l/11 L 11/1'-1
l=k
We must prove that the right-hand side of (10.54) does not exceed w(j + 1)1'/1'-1 I\'ow
J
(10.55 ) L Is: t .Ii:
- 1
fL
PERTURBATIONS FROM SYMMETRY 69
Comparing (10.54) and (10.55), we see that to get (10.47) for] + 1, it suffices that w satisfies
(i) bk ::; w(l 20)(J + 1)1'//1-
1,
(10.56) (ii) (31 ::; wo,
(iii) (31 w1/1' ::; wO /1
for some 8 E (0,1). Since J 2: 2k, (1O.56)(i) holds if
(10.57) 1 ::; (1 - 20)21'/1'-1,
which is certainly satisfied for 0 near 0. With 0 so chosen, (1O.57)(ii) and (iii) also hold if OJ is large enough. Thus we have (10.47) for all k 2: k and the proof is complete.
REMARKS 10.58. (i) If f(x) in (10.2) is replaced by f(x,lL), where
(10.59) If(x,OI::;03+04IElo, 0::;0'</1-1,
and (10.5) is strengthened to
(10.60)
a variant of our above arguments gives a stronger version of Theorem 10.4 for this case.
(ii) It is an interesting open question as to whether Theorem 10.4 holds without (10.5). Bahri [Ba] has given a partial answer. He proved for
- 6:.1L =
1L = 0,
+ f(x),
x E 80,
x EO,
where s < (n + 2)(n - 2)-1, that there is an open dense set of f in W- 1,2(0) for which (10.1) possesses an infinite number of distinct weak solutions.
(iii) A recent announcement of Bahri and P. L. Lions [BLI improves the value of sin (10.5) to s < n(n - 2)-1
11. Variational Methods in Bifurcation Theory
In this final chapter, the usc of variational methods in bifurcation theory will be studied. Recall that if X and Yare Banach spaces, 1: X -+ Y, and 1-1(0) contains a curve Z, we sayan interior point z E Z is a bifurcation point for 1 with respect to Z if every neighborhood of z contains zeros of 1 not on Z. After some preliminary transformations a situation often encountered in applications is X = R x E with E a real Banach space. Y = E. and
(11.1 ) l(A.u)=Lu+H(u) A11.
where A E R. 11 E E. L E [(E. E). and H E CI E) with H(u) as u. ~ O. Then we can take Z == {(A,O)IA E R} or any subinterval thereof. It is easy to that a necessary condition for 0) E Z to be a bifurcation point is that 11 E alL). Indeed if 11 7'- alL), then for A near 11, L - A id is an isomorphism and l(A, u) = 0 is equivalent to
(11.2)
The right-hand side of (11.2) is 0(111111) as 11 -+ O. while 11111111- 1 is of norm 1. Hence {(A,O)IA is near 11} are the only "small" solutions of (11.2) for A near 11 and (11.0) is not a bifurcation point.
Simple examples show this necessary condition is not sufficient. E.g. take E = R2 11 (UI,U2), Lid, and H(11) = (-11~,Ull so l(A,11) 0 becomes
(11.3)
~1ultiplying the first component of (11.3) by U2, the second by 111, and subtracting yields 1(/ + 11~ 0 for any solution (A,U). Hence (11.3) has only trivial solutions. Thus 11 = 1 E alL) but (1. (0.0)) is not a bifurcation point. Surprisingly. hO\\'('wr. if variational structure is present. II E alL) is not only necessary blll is also. modulo some additional technicalities, a sufficient condition for bifurcation to occur. To be more precise, suppose E is a Hilbert space, <li E CI(E, R), and D denotes the duality mapping between E* and E. Then D<li': E -+ E.
THEOREM 11.4. Suppose E is a real Hilbert space and I E C2(E,R) with D<li'(u) = L11 + H(11), where L E [(E, E) is symmetric and H(11) = 0(111111) as
70
VARIATIOI\AL METHODS JI\ B1FTI1CAT!OI\ THEOPY 71
u -+ 0. If J1 E alL) is an Isolated eigenvalue of then (11.0) 18
a bifurcation point for
:().,u)==Dq,'(U)-Al1=Lu+ -AlL.
Aloreover there is an TO > ° such that (il for each I' E (0, ro) there exist at least tll.;O IILstmet so!uI7lln., (,I, If, (r)).
i = L 2, of: = ° having 1:11, rand - 111 small. (ii) As r -,0, (A,(r), 11,(1')) ~ 0).
REMARK 11.5. Actually I need only be defined on a neighborhood of 0 in E. Theorem 11.4 is due independently to Biihme iBii) and .\1arino [Maj. Aside from smoothness considerations, it generalizes an carlirr result of Krasnosrlski [K]. Minimax methods are not needed to prove Theorem 11.4. A standard construction from bifurcation theory converts the problem of finding zeros of : to a finite dimensional problem in R x N(L - J1 idl, where N(Al denotes the null space of a linear operator A. The latter problem will be solved by obtaining the maximum and minimum of a funct.ional defined OIl a sphcrclikc manifold ill N(L - J1id).
The proof of Theorem 11.4 begills with the finite dimensional reduction mentioned above. Let N == N(L J1id) and let NJ. denote its orthogonal comple. ment. Then E = N ttl NJ.. and if 11 E E, 11 == V wEN ttl NJ.. Let P and pJ. denote respectively the orthogonal projectors of E onto N, N 1.. We seek solutions of the equation
(11.6) Lu + H(u) = AU
near (J1,0). Using P and pJ.., (11.6) is equivalent to the pair of equations
(11.7)
(11.8)
Set
Lv+PH(v+W)=AV,
Lw+PJ..H(v+w) = AW.
F(\v,w) == (L-Aid)w+PJ..H(v+w).
The hypotheses on I imply F E Cl(R x N X NJ..,NJ.). Clearly F(J1,O,O) = 0. Let Fw ().L, v, w) denote the Frechet derivative of F with respect to w. Then Fw (J1,O,O) = L - ).Lid, an isomorphism of N1. onto NJ... Therefore by the Implicit Function Theorem, there is a neighborhood, 0, of ().L, 0) in R x N and a mapping ;p E C 1 (0, N1.) such that ;P().L, 0) = ° and the zeros of F near (J1, 0) are given by {(A,v,;p(.A,v))I(A,V) EO}, Thus to solve (11.6) near (J1,0), it suffices to solve the finite dimensional problem
Lv + PH(v + ;P(A, v)) = AV,
or equivalf!ntly since Lv = J1V,
( 11.9)
This reduction is called the Method of Lyapunov·Schmidt.
72 \'AI\IATIO:;AL ~!ETHODS I:; BIFURCATIO:; THEORY
Before continuing further, two useful estimates should be pointed ouL Since L - A id is an isomorphism of j\" ~ to X for all A near ~I, (IL8) can be rewritten
as
(1110)
Since H(A,U) = o(llull) as u·~ 0, (11.10) shm\'s
:PtA, v) = 0(111' + :PtA, 0')11)
iL'i v-" ° imel therefore
(11.11)
ii," V ~ ° uniformly for A near ~l. Differentiating (11.8) with respect to A gives
(1112) (L - Aid, pi- H'(v T :ptA, v))) ~~ (A, v) = 'P(A, v),
Since H'(O) = 0, (lUI) and (IU2) imply
( 11.13)
a.5 l' ~ ° uniformly for A near 11, The next step in the proof of Theorem 11 A is to find A as a function of v near
0). By (11.9), !\ \/1. - il) , V)l
or
( 11.14) A -11 =0
for v I ° and (A, 1') near (11,0), Define G(A, v) to be the left-hand side of (11.14) and G(A,O) == A-/l. The estimates (lUI), (lU3) and properties of H show G is continuous from a neighborhood of (11,0) in R x N to R, G(I1, 0) 0, G is
continuously differentiable in A near (11,0), and also with respect to v for v I 0, Since (JC(/I, O)/OA = 1, a stronger version of the Implicit Function Theorem [Di] implies the zeros of C near (11,0) are given by A = 1jJ( v), where 1/J is continuous in a neighborhood of 0, 1.6(0) 11, and 1/J( v) is continuously differentiable in a
deleted neighborhood of 0.
I"ow set XlV) = 'P(1.6(v),o'), Then X is a continuous map of a neighborhood of ° in N to lv'"- and is continuously differentiable in a deleted neighborhood of 0, Moreover Xlv) = o(llvll) as v -> 0. For our later purposes, the behavior of X' Ilcar 0 must be studied. By (11.8) for zEN and small t E R,
(11.15) (L - + t.z)idJx(v + pi H(l' + tz + Xlv + tz)) = 0,
Diffcn'ntiating (11. 1,5) at t 0 gives
+ p~ H'(l' ~ X(l')))x'(v)z
-pLH'(v't' ( 111G)
(L-
VARIATIO"AL :-'1ETHODS E\ BIFCRCATIOl\ THEORY
Similarly (1114) yields
(?}J'(v). z) = ilvll~21(H'(v + X(v))(z T X'(v)z). v) + (H(v + X(v)),z)]
- 2(H(1I + X(lI)),V)(v, z)llvll~4 (11.17)
Consequently since H'(O) = 0. by (11.16)~(11.17), for v near 0, l' f 0.
Ilx'(v)zll :s: constll(1jJ'(v), z)i"x(V)11 + IIH'(v -i- xlv)):!
( 11.18)
as v -+ 0, v f 0, or
(11.19)
:s: const { + Ilx'(v)zll)
~'-;,-",;---'-'izi'l"x(v)i' -+- x(v))11
= o(l)llzll + o(1)llx'(v)zil
Ilx'(v)zll o(I)llzll
73
as v -+ 0, v f 0. Equation (11.19) shows X'(v) extends continuously to v = 0 and ;((0) = O.
With these preliminaries in hand, the proof of Theorem 11.4 can be reduced to the study of a finite dimensional variational problem. Let 1) be a neighborhood of ° in N in which 1jJ and X are defined and C 1 Let ]v( := {v -i- x(v)lv E 'V}. Then ]v( is a CI manifold of dimension n dim N in E. Let E > ° and set D, := ]v( naB, so for small E, D, is a compact CI, n 1 manifold. Consider liD,. This functional possesses at least two distinct critical points corresponding to the maximum and minimum of I on D,. We claim that at any critical point U of II D"
I'(u) = AU, where u = v -+- xlv) and A = 1jJ(v). Assuming this for the moment, observe that as f -+ 0, if v€ E D€, then v€ -+ O. Therefore v€ -+- x( v,) --+ 0 and 1jJ(v€) -+ fL. Thus the proof of Theorem 11.4 will be complete once the above claim is verified.
If u is a critical point of liD" then
(11.20) J'(u)<p=o
for all <p E T D€u, where T M x denotes the tangent manifold to manifold M at the point x. Now
and T(aB,)x = {<p E EI(<p,x) = O}.
Therefore
TD,x = {<pE T]v(xl(<p,x) = O}.
Equation (11.20) readily implies
(11.21) (DI'(u) - s-2(DJ'(u),u)u,<p) = 0
i4 \'AHIATJO,\AL \!ETHODS 1,\ BlFURCATJO" THEORY
for all p E span{1!.TDcu}' We will show that (1121) yields £'(11) = AlI, where
11 I' + ilnd A = Since v EM implies 11 = l' + xlv). where l' E V, the constrnction of X shows P"- D1'(lI) c= 1.'(1')X(1'), Therefore
( 11,22) (DI'(1J), xlv)) = l.0(1')llx(v)11 2
The constrnction of 1;' further implies
(11.23)
Hccillling t hilt
(11,24)
for 1L E DE' Therefore
(I,'(v) - 11,) 2 = (DH(v),l').
(11.22)(11.23) yield
(DI'(V).l1) = 0(v)iil1I1 2 =
1)(v) = c- 2(DJ'(1l),11) = A(V)
for illl v E DE' Compilring this to (11.21) shows
(11.25) (DJ'(l1) - 1))(1J)11. 'P) = (]
for illl E span{l1. TDE ,,}, By (11,8), PJ'(DI'(1I)'- Ali) = (] so (11.25) gives
(11 ,26) (DI'(l1) =(]
for all 'P E spiln {11, I' DE", ;V J. } == W. We will show W = E and therefore D £' (11) = A 11. To prove this, note
first that 11 = v + X' (11)1' for some v E V and X' (1J) E N 1.. Therefore W =
span{1J, TDEu , N~}. Let VI,.· .• 1Jn -1 be a basis for I'D,,,. To get a basis for I' .Mu, we supplement these vectors by a vector 1Jn , Since TMu = {x + x'(1J)xlx EN}, we have
n
(1127) 1J+x'(V)1J=L 1=1
\'ote that (3n f (] in (11.27) for otherwise, v + X'(v)v E TDeu and therefore
(11.28) (v + X'(v)v, 11) = (] = + (X'(v)v,X(1J)) + 0(livI12)
for v near O. Hence v = 0 if is small, contrary to ilv x(v)11 = E.
Thus (3'1 of O. Solving for 1Jn in (11.27) shows that
( 11.29) 1Jn E span{v, TEu,;V.l.} = w.
Finally observe that if y E E,
y (P + x'(1'lP)y + (P~ - X'(1')Py) E span{T Mu , NJ.}
= span{vn , TD,u, N1.} c W
via (11.29). Therefore W = E and the proof of Theorem 11.4 is complete.
VARIATIONAL METHODS IN BIFURCATION THEORY 75
COROLLARY 11.30. If I is even and dim N ~c n. the equation DI'(I1) AU possesses at least n distinct pairs of solutions (AJ (r) ± iLJ (r)) f 1 S .7 < n. such that !luJ(r)!l = r. Moreover AJ(r) ~)1 as r --+ O.
PROOF. If I is even, 'P and X are odd functions, DE is a symmetric set, and 110, is even. A slight modification of the proof of Theorem 8.1 then gives the corollary. We shall not carry out the details.
We will briefly sketch an application of Theorem 11.4 to a bifurcation problem for a semilinear elliptic partial differential equation. Consider
(11.31) ~ 6u A(a(x)u + p(x, u)), x E 0,
n = 0, x E a~,
where p E C1 (n, R) and satisfies (P3) and a is as in (2.40). Let X(~) E COO(R, R) satisfy X(O = 1 for I~I S 1, xlO = 0 for I~I ;:> 2, and 0 S xW ~ 1 for all ( Define
p(x,~) = X(Op(x, 0 + (1 ~ xlO)· Then p E C1, and satisfies (P2) (with s = 0) and (p3). Set
l(u) r (~an2 + F(x, n)) dx in with 11 E E = Wc;,2 (0) as usual. The properties of p and PropositiJn B.34 imply I E C 2 (E,R). The argument of Theorem 8.17 shows that a critical point of 11aB, satisfies
n))'Pdx = v / Vn· v'Pdx
for some v and all 'P E E, i.e. n is a weak solution of (11.31) (for p) on aBT with A = v-I In terms of Theorem 11.4, using (P3) it is not difficult to check that
H(u)'P = In p(x, n)'P dx ,
L is defined by
= In {LU'P dx,
and alL) = {,u E Rlau ,u6n 0 for some n 't 0, u E E}, i.e. alL) = p;l is an eigenvalue of (2.40)}. Since these eigenvalues are of finite multiplicity and isolated, by Theorem 11.4, each eigenvalue of (2.40) provides a bifurcation point for (11.31) (for pl. Arguments from elliptic regularity theory show if r is small enough, IluIIL=(n) < 1 and therefore p(x, u) = p(x, u). Thus we have shown
THEOREM 11.32. Ifp E C 1(n,R) and satisfies (P3), every eigenvalue A) of (2.40) gives rise to a bifurcation point (AJ' 0) of (1131).
REMARK 11.33. If p is odd, Corollary 11.30 can be used to say more about the number of solutions of (11.31) near (A J' 0).
76 \'ARIATIO,\AL \lETHODS 1:\ BIFURCATIO,\ THEORY
Our final result in this chapter is a variation on Theorem 11.4 where we seek
80111 tions of
(11.34) Lu + = Au
as functions of A rather than of r = Iluli. This is a somewhat morc subtle situation than the case treated in Thcorem 11.4 and involves ideas of a minimax
nature.
THEOREM 11.35 ]R3]. Under the hypotheses of Theorem 11.4, either (i) (fl,O) is not an isolated solution of (11.34) in {Il} x E, or (ii) there is a one sided neighborhood A of 11 such that for all A E A \ {Il},
( 11.34) possesses at least two distinct nontrivial solution.5, or (iii) there is a neighborhood ,\ of fl such that for all A E A \ {ILL (11.34)
possesses at least one nontrivial solution.
PROOF. By the proof of Theorem 11.4, (11.34) is to (11.9) where 1') E ('1 and ,:;(,\, v) = o(li as v ~ 0 uniformly for A ncar 11. Define
g(A, v) == I(v + p(,\, v)) + v)
Then 9 is ('1 for l' neal' 0 in:V. Letting h denote the primitive of H with h(O) 0 shows
(1L36) vi = )0) +h(l'+
\Ye claim nitical points of g(A. ncar v = 0 are solutions of (11.9). Indeed at a
critical point of 9 we have
( 11.37) + v))D + Iw(v + ptA, v))Pv()', v)iJ
- AI( v, D) + (p(A, u), Pv(A, v )iJ)],
Siner by (11.7).
for all Ii.' E .\"-, (1 simplifies to
0= (CJ,.(A, n = (Dlv - AV, D)
for all l' E"Y which is equivalent to (11.9). \'ow ill order to prove Theorem 11.35, we need only analyze the critical points
of 9(A,) near 0 for A near 11 The form of 9 implies v = 0 is a critical point of urA,) for all A near /1. If 0 is not an isolated critical point of g(ll, ,), then alternative (i) of Theorem 11.35 holds. Thus for what follows we assume 0 is an isolated critical point of 9(11, .). Consequently 0 is either (a) a strict local maximllTll or minimum for g(p, .], or (b) 9(11,') takes on both positive and negative \'al11es ncar O. The former possibility leads to alternative Oi) and the latter to alt crnative of Theorem 11.35. Before verifying these statements, note that
VARIATIONAL METHODS IN BIFURCATIOK THEORY 77
g(/1, u) g(A, u)
Case (a)
----~+::__----u
g(A,u)
Case (b)
----~_+_'''-------u ----~~~-----u
for the special case of dim N = 1, one can use the shown figures as a basis for a proof.
Case (a); 0 is a strict local maximum or minimum for g(/1, .). We will assume that 0 is a strict local maximum for g(/1, .). Replace 9 by -g to handle the case of a strict local minimum. Since g(/1,O) = 0, for r > 0 and small, there exists a (3 > 0 such that g(/1,.) ~ -2(3 on BEr . (Here and for the remainder of this section the sets we are working with lie in N.) Therefore for A near /1,
g(A,) ~ -(3 on BEr . By (11.36) and previous remarks,
(11.38) g(,\,v) =
as v -t O. Hence g(A, u) > 0 if A < Ii and 0 I v is sufficiently small. In particular there exists apE (O,r) such that g(A,V) 2: (/1 A)4- 1p2 == a(A) for v E BEp. Let
(11.39) e(A) == max g(,\, v). vEB,
Then for A < /1 and A sufficiently close to /1, e(A) 2: a(A) > 0 and is a critical value of g(,\, .). Moreover g(A,.) satisfies hypotheses (Id and (h) of the Mountain Pass Theorem, the latter with e being any point on BEr . Theorem 2.2 cannot be applied directly since g(A,.) is only defined near 0 in N. However since a(A) > 0 > -(3, it is easy (via the proof of the Deformation Theorem) to
78 VAHIATI00:AL \fETHO[)S 1:\ BlFl:HCATI00: THEORY
constmct a ITlap T) C(:O, I' x B r, Br) such that I/(t,·) = id on BBr and T) has
the properties of Theorcm A.4 rclati\'c to Dr' Consequently if
(11.40)
r={hEC(iO,l],B r ):h(O)=O andh(l) BBr }
lllax UP, h(I)), tEio.1j
the proof of Theorem 2.2 shows <;;(.\) ::: a(.\) > 0 is also a critical value of
1](.\,) for .\ < )1 and near p If <;;(.\) < 7:(.\), it is clear that (11.34) has two
distinct nontrivial solutions If <;;(.\) = e(.\), (11.39)-(11.40) show c(.\) equals the
maximum of a(.\,·) over ever:, curve in 1'. Therefore the set of points in Br at
which a(.\,·) achieves its maximum in Br separates 0 and BBr, Consequently this critical set contains the boundary of a neighborhood of 0 in ;V, In any event
we sec Ca'il' (a) implies altcrnative (ii) of Theoem 11.35, Case (b): g(p,,) takes on both positIve and negative values near 0, This case
is more subtle than the previous one, Again the idea is to reduce the proof to iUl argumcnt of "mountain pass" type but a more complicated construction is
required. To begin, the computation in (11.37) shows
9t,(.\,V) = (fl ~ .\)1' + PH(1' + rp(.\,1'))
which is C1 in (.\, v), Hence the negative gradient flow corresponding to 9(fl., ,):
(11.41) 1,0(0, v) = v,
has a unique (local in t) solution for all v near 0 in ;V, Let Br be a small ball in
which this flow is defined, Set
S+ = {x E Brl1,6(t, x) E Br for all t > O}
and
S- = {x E Brl1,6(t, x) E Br for all t < O},
LEMMA 11.42, S, andS- are nanempty,
PROOF, We will only verify the S+ case, Let (xm) C Br be a sequence such
that Xm -+ ° as m -+ 00 and 9(fl, > 0, Consider 1,6 ( t, By assumption,
9v(J.L,V) f ° in Br for v f 0, If for fixed m, the orbit 1,6(t, remained in Br for all t < 0, it would have to converge to a critical point of 9(fl, ,) with positive
critical value, Since this is impossible, there exists a smallest tm < 0 such that
1,0(tm , = Ym E DBr, Since Xm -+ 0 as m -+ 00 and 9v(J.L,O) 0, tm -+ ~OO
as m -+ 00, Let y be a limit point of (Ym)' Then 1,6(t, y) E Br for all t > 0 so yES+
REMARK 11.43. Since H(u) = a(llull) as u -+ 0, the matrix 9vv(J.L, v) vanishes at v = 0, If it were nonsingular, S+ and S- would be respectively the stable
and unstable manifolds at ° of the flow given by 9v (fL, -).
VARIATIONAL METHODS IN BIFURCATION THEORY 79
PROPOSITION 11.44. There exi"ts a neighborhood Q of 0 in ,V and constants c+ > 0 > c- such that Q c Br and v E aQ implies either
(i) g(f.1.,v) = c+, or (ii) g(f.1., v) = C ,or (iii) 1/;(t, v) E aQ for all t near O.
Aosuming Proposition 11.44 for the moment, we can complete the proof of Case (b). Let T- = g(f.1., .)-I(C-) n S-. The construction of S' and Q shows
T- 7= C. Set
Define
(11.45)
f:={hEC([O,l],Q)lh(O) O,h(1)ET-}.
erA) = inf max g(.\, h(t)). hEr tEla,l]
Letting A < J1 and choosing p as following (11.38) shows c(A) 2: a(A) > O. We claim c(A) is a critical value of g(A, .). Indeed (A.1) shows -gv(f.1., v) is a pseudo
gradient vector for -gv (.\, v) for all v E aQ. This permits a modification of the proof of the Deformation Theorem to find TI E C([O, 11 x Q, Q) having the usual
properties and the use of TI(1,') in the usual fashion to conclude that erA) is a
critical value of g( A, .). If A > f.1., the same argument works on replacing 9 by -g. rt remains to construct the set Q. Let Ac:= {v E Nlg(f.1.,v):S e}.
LEMMA 11.46. There is an c > 0 such that if c := maxvEB, g(f.1., v) and x E BE \ S-, then as t -+ -00, the orbit 1/;(t, x) leaves Ac n Br via Ac \ aBr.
PROOF. If not, there are sequences Cm -+ 0, Xm E BEm \ S-, and tm < 0
such that Zm := 1/;(tm, Xm) E aBr and
min g(p, v) :S g(f.1., zm):S max g(f.1., v). vEBem vEB<>m
Therefore along a subsequence, Zm -+ Z E aB r and g(p, z) O. Let Tm < 0
denote the time of final exit of 1/;(t, xm) from BEl as t -+ -00 and consider
1/;(t,xm) for t E (tm,Tm). Since Ilgv(f.1.,v)11 is bounded away from 0 in 13r \ BEll (11.41) shows that
Om:= 9(fl,Zm) - g(fl,1/;(Tm ,Xm )) 2: const > 0
if Itm - Tml is bounded away from O. But our above remarks show om -+ 0 as m -+ 00 so Itm - Tmi ---t 0 as m -+ 00, i.e. the time it takes the orbit to go from
1/;(Tm' xm) on aBEl to Zm on aBT tends to O. However this cannot happen since gv(fl, v) is bounded away from infinity. Therefore there exists an e as desired.
An obvious consequence of Lemma 11.46 is
COROLLARY 11.47. There is an E > 0 such that if
c- = min g(f.1.,v), vEBi;
80 VARIATIONAL METHODS IN BIFURCATIOCI THEORY
and x E B£ \ 5--, as t -4 -00 the orbit 1jJ(t, x) leaves Ac+ n aBT via Ac+ and if x E Be \ ST, as t -4 00 the orbit 1/J( t, x) leaves Ac ,n BT via Ac .
;-.Jow let E be as given by Corollary 11.47. For each x E BE \ S-, there is a
corresponding t- (x) such that g(l1. (x), x)) = c+ and for each x E Be \ S+. there i8 a tT(x) such that g(I1.1jJ(t+(x),x)) = c. If x E S+, set t-(x) = -00: if
x E 5-. set t-(x) = 00. Finally define
Q == {'0(t,x)lx E Be and t-(x) < t < t+(x)}.
PROOF OF PROPOSlTIOl\ 11.44. Clearly Be C Q. If Z E Q \ Be, Z = '0(t, x) for some x E Be and t E (t-(x),t+(x)). Therefore for 0 small, 1/J(t,BIi(x)) c Q and is a neighborhood of z so Q is a neighborhood of O. Let v E aQ and suppose
g(l1,v) cI cT or C-. Consider Ov == {1/J(t,v)lt E R}. Since v = limm~(X)vm,
where Vm E Q, and 0,,= n Be cI 0, there exists an x E Ov n BE' Therefore .T= .1')forsomeTERandv= where-TE (x),t+(x)) since
9(11.,11) E (c-. ). Hence x E aBE for otherwise v E Q. Moreover 9(11, 1/J(s, v)) E ,cT
) for s near O. Since v) = '0(s - T,X), it is clear that 1/J(s, v) E DQ. Indeed 1iJ( s, v) = limm~oo 1jJ(s T, where Xm E Be, Xm -4 x, and therefore s - T E (t- (x m ), t+ (x m )). The proof is complete.
REMARK 11.48. If I is even, a much stronger result than Theorem 11.35 holds
(see iFR]).
Appendix A
The goal of Appendixes A and B is to prove some results that were required earlier which are rather technical in nature. In particular in this appendix we will prove a fairly general version of the Deformation Theorem. Thus let E be a real Banach space, U c E, and I E Cl (U, R). Then vEE is called a pseudo-gradient vector for I at u E U [P2] if:
(A.l) (i) Ilvll::; 211J'(u)ll,
(ii) J'(u)v ~ 11J'(u)112
In the future pseudo-gradient will be denoted by p.g. for short. :\ote that a p.g. vector is not unique in general and any convex combination of p.g. vectors for I at u is also a p.g. vector for 1 at u.
Let I E Cl (E, R) and E == {u E EII'(u) Ie a}. Then \f: E -~ E is called a p.g. vector field on E if \f is locally Lipschit~ continuous and \fIx) is a p.g. vector for I for all x E E.
LEMMA A. 2. If IE Cl (E, R), there exists a p.g. vector field for I on E.
PROOF. For each u E E, we can find a vector 11J E E such that = 1 and J'(u)11J > ~III'(u)ll. Then z = ~III'(u)ll11J is a p.g. vector for I at u with strict inequality in (i) and (ii) of (A.l). The continuity of I' then shows z is a p.g. vector for all v E Nu, an open neighborhood of u. Since {Nulu E E} is an open covering of E, it possesses a locally finite refinement which will be denoted by {MJ}. Let PJ(x) denote the distance from x to the complement of MJ. Then PJ(x) is Lipschitz continuous and PJ(x) = 0 if x f/:. MJ. Set
/
/3J(x) = PJ(x) / I:>k(X). k
The denominator of /3) is only a finite sum since each x E E belongs to only finitely many sets Mk . Each of the sets MJ lies in some Let z) == ~liF(uJ)II11JJ' a p.g. vector for I in M) and set \fix) == Li (x). Since o ::; /3J(x) ::; 1 and L J /3J(x) = 1, for each x E E, \fix) is 'a convex combination of p.g. vectors for I at x. Moreover \f is locally Lipschitz continuous. The proof is complete.
81
82 APPE:\DIX A
COROLLARY :\.3. If I(x) 1S even in x, I has a p.g. vector field on j;; given by an odd W.
PROOF. Suppose I is eVeIl. Let \; be as given Lemma A.2 and set W(x) =
~ (V (x) ~ V ( . Then 11-' is odd. locally Lipschitz continuous, and since l' (x) is odd in T.
(i) ::; 1( + I!. (ii) = V(x)\/(x) ~ 2: Recall that A.., =' {11 E ::: s} and Ie == {11 E EII(u) sand 1'(11) = O}.
"ow we will prove the following version of the Deformation Theorem (which in particular contains Proposition 2.1 as a very special case). See e.g. [Brl, C, P1, Sl] for earlier such results.
THEOREM A.4. Let E be a real Banach space and let I E CI(E,R) and satisfy (P8). If C E R, E > 0, and 0 is any neighborhood of K e , then there exists an c: E (0, E) and T) E 0([0.1] x E, E) such that
10 T)(0.11) = 11 for all 11 E E. 2° T)(I.u) = 11 for all tE [0,1) if 1(11) t/: [c E.e +E]. 3° T)(t.11) is a horneornorphl:srn of E onto E for each t E [0, I]. 4° !1T)(t.u) ~ uli ::: 1 for all t E [0,1] and u E E. 5° I(T)(t. ::: I(u) for nil t E [0,1] nnd u E E. 6° T)(l,A o +£ \ 0) C ..10 -£,
7° If Kc = Z, T)(1, C ..10 -£
8° If I(u) is even in 11, T)(t, u) is odd in u.
PROOF. The function T) will be constructed as the solution of a suitably modified negative gradient flow for I. A few preliminaries are needed before setting up this differential equation. By (PS), Kc is compact. Set Ns == {11 E
Eiliu ~ Kc II < t5}, where liu ~ Ko II denotes the distance from 11 to Ke. Choosing (j suitably small, Ills cO. Therefore it suffices to prove 6° with 0 replaced by Ns. ]\'ote also that if Kc = Ns == Z so we get 7° instead.
We claim there are constants b, s > ° such that
(A.5) 1I1'(u)11 2: b for all u EAcH \ (Ae-i U No/8 )·
If not, there are sequences bn -> 0, sn -> 0, and 11n E AeHn \ (Ae-in UNo/8 ) such that Il1'(un)11 < bn . By (PS), a subsequence of Un converges to u E Kc \ No/s· But this latter set is empty. Hence there are constants b, s as in (A.5). Since (A.5) still holds if E is decreased, it can further be assumed that
(A.6) o<s<min(E,~~,~,D. Choosing any c: E (0, s), define
A == {u E EII(u)::: c ~ E} U {u E EII(u) 2: c + E}
and B == {u E Elc ~ E ::: I(u) ::: c + c:}.
APPEN])]X ,\
Therefore All B = 0. Set
Then 9 = ° on A, 9 = 1 on B. ° ~ 9 ~ 1, and 9 is Lipschitz continuous on E. Similarly there is a Lipschitz continuous function f on E such that f = 1 on
E \ Nb/4 , f = ° on No/s , and ° ~ f ~ 1. Note that if 1 is even, A, D, and Nb will be symmetric sets with respect t.o the origin and f and 9 arc even functions. Nextdefineh(s)=lifsE[O,ll andh(s) = 1!8if.s2 1. Since [ C1(F::,R).by Lemma A.2 and Corollary A.3, there exists a p.g. vector field l/ for [ on E with V odd if 1 is even. Finally set W(x) = -f(x)g(x)h(lIV(x)II)ll(x) for x E E and W (x) ° otherwise. Then by construction, W is locally Lipschitz continuous on E with ° ~ IIW(x)11 ~ 1 and W is odd if 1 is even.
Now we can define the mapping 1/. Consider the Cauchy problem:
d1/ (A.7) dt W(1/), 1/(O,u) = u.
The basic existence-uniqueness theorem for ordinary differential equations implies that for each u E E, (A.7) has a unique solution defined for t in a maximal interval (u), t+(u)). We claim t±(u) ±oo. If not, say t+(u) < 00. Let tn -> t+(u) with tn < t+(u). Integrating (A.7) shows
(A.S) - tn !
since !IW(·) II ~ 1. But then 1/(tn, u) is a Cauchy sequence and hence converges to some u as tn -> t+(u). The solution to (A.7) with u as initial data then furnishes a continuation of 1/( t, u) to values of t > t+ (u) contradicting the maximality of t+(u). Similarly t-(u) = -00,
The continuous dependence of solutions of (A.7) on the initial data u implies 1/ E C([O, 11 x E, E) and (A.7) implies 10 holds. Since E > E, g(x) = ° on A so 20
is satisfied. The semigroup property for solutions of (A. 7) gives 3°. Integrating (A.7) and using IIWC)II ~ 1 and 10 gives 40
. Above remarks on the oddness of W when 1 is even yield So. To verify 50, note first that if W(u) = 0, 1/(t, u) == u is the solution of (A.7) (via uniqueness) so 5° is trivially satisfied. If W(u) f= 0, u E E so V(u) is defined as is V(1/(t,u)) and
~~...!..!.. = 1'(1/(t,u))~: (A.9) -1'(1/(t, u))f(1/(t, u))g(1/(t, u))
x h(IIV(1/(t, u)!!)ll(1/(t, u)) ~ ° via (ii) of (A.l) and 5° follows for this case.
It remains only to verify 6° or a fortiori, 1/( 1, .10 +£ \ Ns) C .10 -£, If u E .1c-"
then 1(1/(t, u)) ~ C-f by 5°. Thus we need only proveu E Y == .1C+E \(.1c_,UNs) implies that 1/(l,u) E .1c - E ' Let u E Y. The reasoning that led to 5° showed
(A.lO) d1(1/(t, u)) < ° dt -'
84 APPE:\DIX A
Since 9 = ° Oil A c - c. the orbit 1)(1. u) cannot enter A c-€. Therefore 10) implies
(A.11) 1(1)(0. u)) I(I)(t, u)) 'S E T E: < 2c'
for all t. 2 0. Suppose that u E }' and 1)(£.u) E Z == Ac+€ \ U for
s E [0, tj. This will certainly be the case for small I. Then for such s, 1)(3. u) E i; (\'ia (A.5)) anel f(7)(S, u)) = 1 = 9(1)(8, u)). By (A.11).
IA.12)
2E 2 fO -I'(rl(.,.1I))h(IIF(1)(s.l1))IIW(1)(s,l1))d,~
= r h(!W(I)(s, u))II)['(1)(s. 11))II(1)(s, u)) ds ./0 2 {h(IW(I)('S, u))ill'M'(1)(s, u))11 2 ds
. ()
2 b 11 h(
b j'l > -- 2 0
b 1t > -- 2 , 0
b
2
u))rI)!II'(1)(,', 11))11 dx
8.
W(1)(s,u))ds b 2
ds
u) - ul!,
where we ,;u(wssively nseo (A.l)(ii). (A.5). and (A.l)(i). Hence by (A. ano
(A.6).
l ) " 4f S 111) t. u - l1i! 'S b < '8
Thus the orbit 1)(t,11) cannot leave Z by entering NO/2' Consequently the only way 1)( t. u) can leave Z is to enter Ac _.[· We claim this occurs for some t E (0,1) thereby proving 6°. If not. 1)(t, u) E Z for all t E (0,1) and as in (A.12),
(.'1.13) -.-'-'--'-_.:.-'.. 'S -hi IW (1) (1, ullli) Ill' (1)( t, u)) 112
If for sOlIle t E (0.1). IW(1)(t. u)) II 'S 1. hlW (1)(t, 11)) II) = 1 and (A.5) and (A.13) imply
(A.14)
On the other hand, if for some t E (0,1), W(r/(t. u)h' 1 so (A.l)(i) and (A.13) yield
(A.15)
COf],;cql1(,nth' for all t E (0. I). we have
(t,l1m > 1, hlIW(1)(t. u))II) =
( A.Hi) dI(1) ( t. u))
( 2 1) - mill \b '=I .
APP~~"D1X A 8.)
Integrating (A.16l and combining the result with (A.1ll gives
which is contrary to (A.6). The proof of Theorem A.4 is complete. REMARKS A.17. (i) :--Jote that the full strength of (PS) was not required in the
above proof. We only needed (PS) to conclude that Kc was compact and that (A.5) holds for some b, E > O. But we can also achieve this by merely assuming (PS)loc: there is a 6 > 0 such that IJ(1171) - ci < Sand J'(lLn) -40 as n 00 implies that (1171) is precompaet. Still weaker forms of (PS) have been found (see e.g. ICe, Chi]).
(ii) It is sometimes useful to have a bit more flexibility ill the definition of a p.g. vector. If (A. 1) is replaced by
(A.l') (il Ilvll:S o:llI'(l1)II, (ii) J'(l1)V ~ 1!2,
where 0 < {3 < a, it is easy to check that Lemma A.2 holds with 11' chosell so that J'(u)w > 2{3(a+{3l-11lw!! and z = l\iI'(u)w!!. Moreover the proof of Theorem A.4 also is essentially unchanged aside from replacing (A.6) by
(3bS 62
16a' 2' (A.6') 0< f < min
(iii) There arc problems where instead of rl(t, u) satisfying 5c 7c well as the other conclusions of Theorem A.4), one wants a function ~(L u) to satisfy J(~(l,u)) ~ flu), and if
As := {u E EIJ(u) ~ s},
then ~(l,Ac_" \ 0) c AC+E (with 0 0 if Kc 0). Such an ~ can be obtained by slightly modifying the proof just given, in particular using a positive rather than a negative gradient flow.
(iv) There are versions of Theorem A.4 for a functional J defined on a Banach manifold, M, rather than a Banach space [Brl, Pl-2, 81-21. Using the same sort of ideas as in the proof of Theorem A.4, and in particular constructing a p.g. vector field for M which is tangential to M, we can get an analogue of Theorem A.4 for such a setting. E.g. in Chapter 7 we consider the case of M a sphere, say oBJ[O), in a Hilbert space. For this special case, a tangential p.g. vector field can be constructed via V(x) - (v(x) . where V(x) is it p.g. vector field for Jon E.
The next result concerns the effect on the mapping 7)(1, u) of making stronger assumptions on the form of I. Such information is useful in particular for highly indefinite functionals such as arise in treating Hamiltonian systems as ill Chapter 6.
APPS:\DIX A
PHOPO:'ITIO:\ A.18. Suppose E 1" (L rcal Hilbert space and IE ('I(E.R) w1th 1 (II) ~ ~: L Ii. Il) -~ ), L 18 selfad)oml. and :;:1 IS compar-t. Thrn
1](1.11) =
lL'here 0 <; 0ll.ll) <; 1 orld 1\: [0,1] x E -" E 18 compact.
PROOF. B~' the proof of Theorem A.4, 'I is determined as the solution of the
initial \'alnc problem
(A.19) 1](0, u) u,
wherr ° < .0' <; 1 and Ii is a p.g. \'ector field for l' on E. Actually since
.0(0 = J(~)!i(Oh(Y(~);I) and f(09(0 = 0 if 1(0 rf:. [c-i,c-i-i] or ifu E SIi/8, j' need onl~' be a p.g. vector field on S =: {~ E E11(0 E Ie i,c + i] ane! E r1 Sli/d. We claim such a F can be chosen so that V(u) Lll W(u) where IF is cOlllpact. Assuming this for the momcnt, (A.19) bccoIlles
(A.20)
Considering the 1] in the argument of wand W as being known, 1] satisfies an
inhomogeneous linear equation ane! therefore it can be represented as
(:\.21 ) 11) = exp (([ u)) dS) L) 11 + K(I, u),
where
K(t, 1/) = - [ [exp ([ ,u)) ciT.
To see that 1\: iO, 1 i x E ~ E is compact, suppose BeE is bounded. \Vith
out loss of generality, B = BR for some R > 0. By 4° of Proposition A.4,
IICO 1] x B FI ) C BH~l Therefore W(/JUO, 1: x Bit)) C W(BR-i·d W(BR-,-l! which is compact. Let
Y=: :3 E [0,11, Z E W(Bltf-d}·
Since the map ],Z) ......,
is a continuous function on the compact set [0, x W(B It~ 1), its range Y is compact. Therefore the closed convex hull Y' of Y is also compact. Now recalling
that ° <; 0 <; L for each T E ro, 1] and u E BR ,
=: exp [([ W(l)(S,U))dS) L] W(l)(T,U))W(7)(T,U)) E Y.
Hence for t E [0,1]'
l ZdTE Y
It follows that K is compact.
APPENDIX A 87
To complete the proof of Propositioll A.IX. we must shOl\ it is possible to choose a p.g. vector field V for l' on S such that F(u) = Lv vF(u). where W is compact. As a first step w(' have:
PROPOS1TIO)I A. 23. Let E be (! illibert space lind 1': E··· E be compo.ct .. Then given any ~i > O. there exists aT: E ~ E such that T is compact. locally Lipschitz continuous. and
(A.24) IIT(u) T(l1)II::;~:
foralll1EE.
Assuming Proposition A.23 for the moment, let, = b/2, where b was defined in (A.5), andlct W be the corresponding mapping given by Proposition A.23 with 1'(11) = p'(l1). We claim F(l1) == Ll1 + W(l1) is a p.g. vector field on S. It is certil.inly IOCillly Lipschitz continuous. We will verify (A. 1') for F with n = 2, (3 = 1. and 11 E S. By Proposition A.2:). (A.5), and om choicc of ,.
W(l1)11 = IILl1 + W(l1)11 ::; IILl1 + p'(ll)11 + IIp'(ll)·· W(11)11
:; 111'(11)11 +,:; 111'(11)11 + b::; 211 1'(11)11.
Therefore (A.l')(i) holds. Similarly,
1'(l1)V(l1) = 1'(l1)(L11 + p'(l1) + W(l1) - p'(11))
:: /11'(11)112 -II1'(l1)lh:: /11'(11)11 2 - ~II1'(11)112 = ~li1'(11)112
Thus we have (A.I')(ii). Finally we can give the PROOF OF PROPOSlTIO;.J A.23. (This result is essentially due to Bend
[BeZ].) First we construct a special partition of unity for E and then use it to
construct T. For each 11 E E. set Su == E l (11) n {v E Ej 111'(11) - T(v)11 < I}' Then {Su 111 E E} is an open covering of E and therefore it possesses a locally finite refinement {Yd. Let p,(11) denote the distance from u to the complement of Yi and
3,(11) == Pi (11) / L PJ(11). J
Then p,(l1) is a (locally Lipschitz continuous) partition of unity as in the proof of Lemma A.2. By construction for each Y" there is a w, E E such tbat Yi C Sw,. Therefore
- T(w,}, v) <
for all 11 E Y, and vEE. Since p,(l1) :: 0 and equals 0 ifu r:f Y,.
Pi(l1)(T(l1) - T(Wi), 11) :; iP,(l1)
for all 11. vEE. Summing over i yields
88 APPEi\DIX A
for all u, vEE. Setting T(u) == L,8,(11)T(w,), it is clear that T is locally
Lipschitz continuous and satisfies (A.24). Moreover T is compact. Indeed let BeE be bounded. Without loss of generality, E == ER(O). Let u E BR(O). Then == L, j3,(u)T(w,). If u rf Y,. 8,(u) == 0 while if u E Y
" u E
and therefore Ilu - < 1. Thus the nonzero contribution to T(a) comes from those terms in the sum where w, E E R +1 · Consequently T(u) is a convex combination of terms in T(BR+1)' Since the convex hull of a compact set is compact, T is a compact mapping.
Appendix B
This final appendix treats the question of verifying in applications some of the hypotheses that are required in the abstract theorems presented earlier. In particular sufficient conditions will be given which insure that certain classes of functionals belong to C 1 (E, R). Furthermore the verification of the Palais-Smale condition will be simplified for these classes.
We begin with a useful technical result.
PROPOSITION B.!. Let 0 c Rn be a bounded domain and let 9 satisfy (gd 9 E C(O x R, R), and (g2) there are constants r, S :2: 1 and 01, a2 :2: ° such that
for all x E 0, ~ E R. Then the map <p(x) -; g(x,<p(x)) belongs to C(LT(O),P(O)).
PROOF. If u E U(O),
In Ig(x, u(x))IS dx :s: In (a1 a2IuI T/5)8 dx
:s: a3 r (J + lun dx JrJ
(B.2)
which shows that g: U(O) -; prO). To prove the continuity of this map, observe that it is continuous at <p if and only if f(x,z(x)) = g(x,z(x) + 'P(x)) g(x,'P(x)) is continuous at z = 0. Therefore we can assume 'P = 0 and g(x,O) 0. Let E > O. We claim there is a 8 > ° such that ilullu(o) :s: 8 implies
Ilg(',u)IIL'(rJ) :s: E. By (gtl and g(x,O) = 0, given any f > 0, there is a 6> ° such that Ig(x, 01 :s: f if xC 0 and I~I :s: 6. Let u E U(O) with IluIIL'(O) :s: 8, 8 being free for now, and set
0 1 == {x E Ollu(x)1 :s: 6}
Therefore
(B.3)
89
90 APPE\,DIX 13
where ill1 i dmotes the mca$ure of Ill, etc. Choo'iC t so that (El" This determines r Let O2 = n \ \/1. Then as in (13.2),
(134)
(D.5) f,' > r .In,
I'dx
or 1\121 :S ('\3 1), Combining (134)(11.5) gives
(E.5) r 19(x, l1(x)JI' deS :::: a, (1 + .In, Choose {; so that a3(1 + o-rw :S ([/2)5, Thus (13.3) and (13.5) imply
119 (- , 11 ) II L' (n) :S [ if i i ull u (n) :S Ii
and the proof is complete.
:S (0:/2)'.
The next result is a standard version of the Sobolev Embedding Theorem. The space W~"(\l) was defined in Chapter 1 and its norm will be denoted by !:
PROPOSITIO'\ 13.7. Let n be a bounded domain m Rn whose boundary is a srnooth man1jolrL 1/71 E ~'V({,2 then 11 E L2n/n n 3) and there
i8 a constant c > 0 SlLe h that
(13.8) u(n) :S ellull for all t E [L2n(n - 2)-1] and for all u E w~·2(n). Moreover the embedding map w~·2(n) ~ £I(n), U --. 1L is compact for t E [1, 2n(n - 2)-1).
PROOF. See e.g. [N! or F]. REylARK B.9. If n = 1 or 2, stronger statements can be made. In particular
(13.8) holds for all t E [1, Xl] for n = 1 while for n = 2 it is valid for all t E [1,(0) with c depending on t.
"'ow a criterion can be given showing that an important class of functionals which arise in studying second order semilinear elliptic boundary value problems belong to C1(W~,2(n),R), (A related result is given in [BR].)
PROPOSITION B.10. Let n be a bounded domain in Rn whose boundary is a smooth manifold. Let p satisfy
(pd p(x.O E x R, R), and (P2) there are constants a 1, a2 > 0 such that
Ip(x, ~)i :S al + a2i~ls, where 0 :S s < (n + 2) (n - 2 t 1 and n 2: 3 If
P(x,O foE. p(x, t) dt
and
(B.ll)
(B.12)
APPE:'-iDIX B
I(v) == r (~ In - P(x, v)) dx.
I'(v)<P= (Cvu,v'P-p(x,u)'P)dx In for all 'P E E == Wd,2(0). Moreover
(13.13) J(U) == in P(x, u(x)) lix
is weakly continuous and J' (u) IS compact.
91
PROOF. By Proposition B.7, (pd and (P2), I is defined on E and the same is true for I' (u) with the aid of the Holder inequality. It is clear that the first term in I is C 1 (even COO) and its Fn?chet derivative is the first term in I'. Hence we must show
J(u) == in P(x,u(x))dx
belongs to Cl (E, R). This will be accomplished in two steps: First showing that J is Fnkhet differentiable on E (and therefore is Lipschitz continuous) and then proving that J'(u) is continuous.
To begin, let u, 'P E E. We claim: given allY E > 0, there exists it [; ~ u) such that
(B.14)
provided that II'PII ::; 8. Therefore J is Frechet differentiable at u and J' is given by the appropriate term in (B.12). In fact the argument given below shows more: 8 = 8(s, Ilull) and hence J is uniformly differentiable on bounded sets in the sense of Krasnoselski [K I.
Setting
we have
(B.15)
Define
w == IP(x, v(x) + 'P(x)) - P(x, u(x)) - p(x, v(x))'P(x)l,
J(u) - { p(x,u)'P In 0 1 == {x E 111lv(xlj 2: p},
O2 == {x E 1111'P(x)1 2: 'I},
::; in W dx.
0 3 == {x E 111Iv(x)1 ::; p and 1'P(x)1 ::; 'I}
with p and 'I free for the moment. Therefore
(B.16) in W dx ::; ~ in. W dx.
92 APPENDIX B
By the ~1ean \'alue Theorem.
(B.17) P(x. (+ 71) - P(x. 0 = p(x. (+ e7lh where e E (0.1). Therefore by (B.I7), (P2). and the Holder in quality.
f IP(x.u(x) -I- 'P(x)) - P(x.l1(x))1 dx Jo, (JUS) < f -I- odll1(T)1 T !'P(x)WI'P(J;)11 d;r .. .In,
::; 0110 1 '(0)
+ 031 0 11 1/"[11111It,·q(0) + 11'Pllt'+'(o)III'PliLon;n-'(O).
where
(LU9) 1 .) n - 2 -+ +--=1. (J 5 2n
O/),ern' that s < (71 -L 2)(- - 2)-1 implies that + 1)-1 + (n - 2)(2n)-1 < 1 ';0 there exists a (J > 1 satisfying (B.19) ,md justifying (E.IS). Combining (E.S) and (B. IS) shows
dx iB.20)
+ Similarly
B)' (B.8) and the Holder inequality,
(B.22) 11111: 2' 06Ijl1llu(fl) 2' 06111111u(Od :::> 06 310 111/ 2
Therefore
where "'viI, "'vi2 - 0 as /3 -., 00. Combining (B.1S)-(B.23) yields
(13.21) /' IJt dx ::; 07[kh -1- }vf1(lluII S + il'PIIS)III'PII· .In, \\'r can assume 6 ::; 1. Further choose 3 so large that °+'112 + M1(llull S + 1)1 ::; o,-':l. Hellcr
(B.25) f IJtdJ;<~ In, - 3
APPENDIX B 93
Similar estimates to those above show (B.26)
r iii dx 5 a3 l [1 + (lu(x)1 + 1'P(x)WI'P(x)ll dx in2 in, 5 a4 (l
J2 [1 + (lu(x)1 + l<p(x)iJ'I(s+I)/$ dX) 8/(s+l) 11'PIIL'c) (il,)
(
, " )"'-(S+I) )1/8+1 <: ILs(1 + Ilull' + Hpl/') L,'p(x) 1 C'C~X)l d:r ,
where m = 2n(n - 2)-1 > s + L Therefore
(B.27) r iii dx 5 a6,(s-t-l-m)!(s+I)(1 + Ilul!S + II'PII S) in, 1)
t\'ext since P E ['1 (0 x R, R), given any E, & > 0, there exists a ::.; = 0) such that
(B.2S) lP(x, ~ + h) - P(x, 0 - Ohl 5 Eihi whenever x E 0, I~i 5 il, and ihl 5 :y In particular if ;] == (3 and ~i 5 (B.S) and (B.2S) imply:
(B.29) dx 5
Choose E so that 3a7E 5 E. This determines ~i. Choose, = (B.16), (B.25), (B.27), and (B,29) yields
(B,30) 10 iii dx 5 2; II'PII + (1 + HullS +llpi!S)
Finally choose 5 so small that
(B.31) a6,I-m/(s+I)(2 + Ilull s )5(m/(8+1))-1 5 ~
thereby obtaining (B,14),
Combining
I)
To prove that J' (u) is continuous, let Urn -+ U in E. Then 11m. ~ 11 in LS+ 1 (0) via (B,S). By the Holder inequality and (B.S) again,
IIJ'(um ) - J'(ulll sup II r (p(x,urn(x)) p(x,'P(x)))p(x)dxi (B.32) Ii<pll:ol in I
ip(x, 01 5 al + a21~IC<8/Q for any or :2: 1 and all x E 0, ~ E R. Hence Proposition B,l implies p E C(L"8(0),L"(0)), Choosing or = (s + l)S-I, we see the right-hand side of (B,32) tends to 0 as m -+ 00 and J' is continuous,
Finally to prove that J is weakly continuous, let Urn converge weakly to 11 in E. Then by Proposition B.7, Urn converges to U in 1'+1(0) since s+ 1 < 2n(n-2)-1
9-1 APPE:\DIX B
\'ia (P2). ('onsequenth' Proposition D.l implies J(l1m) ~ J(u). Since J is
\I'eakly continml1ls and .J' is ulliformh' differentiable on bounded subsets of E. an abSlract theorelll X implies .11 i,.; compilrt. (Alternativply, let be
boundr:d in E. Then illon:-; a subsequence. 11m COll\'ergcs wrakly to some 11 E E ilnd lim ~ 11 ill U' 1 (ll) via Proposition I3. 7. The proof then concludes via
rB_32) illlClthe ,.;('Ilt<:llces which follow it.)
RDIARK D.33. The ideas that were used in the proof of Proposition 8.10 call be emplo~'ed to prove related regularity results. E.?!;.:
PHOPO"ITIOX D.34_ Let n be a bounded domaIn In R n whose boundary 18
II smooth manifold. Let p satisfy ())I) p E elm x R.R), and
(P2) there aTC constants al.1l2 > 0 such that
wherr;() .,«lIt2)(rr-2)-lllndrr2'3. If
and
P(x,O == !'~ t) dt .0
I(u) == l (~IVll,2 - P(r 11 dr .0
then 1 ('2(\l'(;2(1l).R). l' 1S (Jiven liy (I3.12), and
r(u)(;:. v) --c j' ('I;. '1t'io all;. t' W~2(1l).
,) dr
The next result concerns the verification of (PS). Recall that (P3) requires any
sequence ) such that (i) l(11m) is bounded and (ii) 1'(11m) --+ 0 as m --+ 00 to be precompact. According to the following result in the setting of functionals
like (B,l1), to get (PS), it suffices to show that (i) and (ii) imply (11m) is a
bounded sequence.
PROPOSITIOi\ B.35. Let p satisfy (pd-(P2) and let I be defined by (B.ll). If ) lS a bounded sequence in E == WJ2 (ll) such that I' ) ~ 0 as m ~ 00, then (u m ) has a convergent subsequence.
PROOF. Let D: E ~ E' denote the duality map between E and its duaL
Then for /1, ;: E E.
(Du):p j' VU' vrpdx, In Thus
(D.36)
APPE\;DlX B 9
Due to the form of this map, tlw conclusioll of the proposition follows if we ca
show J'(urn) has a convergent subsequcllce. Indeed the continuity of D I an,
(B.36) then imply
hm=D-IJ' ) + D- I JI(l1",) -4 lim f) I
the limit being t.aken along the convergent subsequencI' of )1(lIm)' But sinc (urn) is bounded in E and by Proposition B1O, JI is compact, J'(u m) has;
convergent subsequence and the proof is complerc.
In Chapter 6 which studies the existence of periodic solnt ions of Hamilto
nian systems, an analogue of Proposition B.lO is needed. The space E =: WI/2,2(Sl,R2n) is defined in Chapter 6, For this space we have
PROPOSITION B. 3 7. Let H E Cl (R2n, R) and satisfy
(B,38)
for some s E [1,00) and for all (E R2n. Then
(B,39) J(z) fo21r H(z(t)) dt E CI(E, R)
with
JI(Z)W = fo21r Hz(z(t))w(t) dt
for wEE, Moreover JI(Z) is compact,
PROOF, Given Proposition 6,6 and (B.38) the proof is almost identical te
that of Proposition B.IO, We therefore omit the details (see also [BR]).
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