Minimization problems for the exterior domain

Calc. Var. 5, 463–487 (1997)c© Springer-Verlag 1997

Minimization problems for the exterior domain

Orlando Lopes

IMECC-UNICAMP, C.P. 6065 13081-970, Campinas, SP, Brazil; e-mail: [email protected]

Received January 19, 1996 / Accepted October 28, 1996

This paper is dedicated to the memory of Mario Barone Junior and Marcio HeleneBarone

Abstract. In this paper we discuss the existence of minimizers for certain vari-ational problems defined by integrals on the exterior of a bounded domain. Weindicate some applications to the existence and stability of solitary waves forcertain evolution equations.

Notation.

1) N ≥ 2 andK ≥ 1 are integers andΩ ⊂ IRN is an open set with boundedcomplementΩc and smooth boundary∂Ω;

2) for 1 ≤ p < +∞ and an integer m,Lp(Ω), Wm,p(Ω), H m(Ω) are the usualSobolev spaces of real valued functions;

3) LpK (Ω) = Lp(Ω)× · · · × Lp(Ω) (K times);

4) Wm,pK (Ω) = Wm,p(Ω)× · · · × Wm,p(Ω) (K times);

5) H mK (Ω) = H m(Ω)× · · · × H m(Ω) (K times);

6) if u, v ∈ IRK then〈u, v〉 =K∑

i =1

ui vi ;

7) if u, v ∈ L2K (Ω) then〈u, v〉 =

K∑i =1

∫Ω

ui (x)vi (x)dx;

8) if u, v ∈ H 1K (Ω) then

〈gradu(x), gradv(x)〉 =K∑

i =1

〈gradui (x), gradvi (x)〉

464 O. Lopes

and

|gradu(x)|2 =K∑

i =1

|gradui (x)|2.

I. Introduction

In this paper we consider the question of existence of global minimizers for

V (u)=12

∫Ω

|gradu(x)|2dx +∫Ω

F (u(x))dx(I.1)

subject to

I (u)=∫Ω

G(u(x))dx = λ > 0,(I.2)

whereF (u) and G(u) are given functions ofu ∈ RK andΩ is the complementof a bounded domain inIRN . For eachx ∈ Ω, u(x) is a vector inRK and theminimization will take place inH 1

K (Ω) and so, the Euler-Lagrange equation fora minimizer is an elliptic system with Neumann boundary condition.

Due to the lack of compactness, besides vanishing, the difficulty is the pos-sibility of occurrence of dichotomy for minimizing sequences.

A very powerful method called concentration-compactness has been devel-oped by P.L. Lions ([1]) to overcome that difficulty. The main result of themethod is that the precompactness (except for translation whenever is the case)of minimizing sequences is equivalent to a certain strict inequality (strict subad-divity) and this has been used in many concrete situations. In [2] that methodhas been used by M. Esteban to prove the existence of minimizer for I.1-I.2 forcertain nonlinearities.

For some minimization problems with just one constraint (which is the caseof the problem above) we have offered a different presentation in [3] and [4](concentration-compactness works also in the presence of many constraints).According to our approach, in order to show the precompactness of minimizingsequences (except for translation whenever is the case), we have to verify acertain crucial condition; for problem I.1-I.2 this crucial condition is assumptionH6 stated in part II of this paper.

Although we use the machinery of concentration-compactnes, if we can verifythat assumption then we can prove the existence of a minimizer without goingthrough the verification of the strict subaddivity.

If Ω = IRN we have shown that this crucial assumption is always satisfied andthen we can prove the existence of minimizers for very general nonlinearities.If Ω is the complement of a bounded set, it is not clear whether assumptionH6

is always satisfied. However it is satisfied ifΩ is the exterior of a ball (for anynonlinearitiesF (u) andG(u)); it is also satisfied (for anyΩ) providedF (u) andG(u) are sum of homogeneous functions with certain restrictions on the sign ofthem. So, at least for the exterior of a ball, we can deal with cases for which

Minimization problems for the exterior domain 465

the verification of the strict subaddivity could be difficult (but not impossiblebecause it is also a necessary condition).

As a possible application we consider the existence and stability of solitarywaves for certain evolution equations. In the caseΩ is the exterior of a ballthis question has been studied by M. Esteban and W. Strauss ([5]) for particularnonlinearities. Here we follow the method introduced by T. Cazenave and P.L.Lions ([6]) which is a combination of minimization technique with Liapunovfunctionals.

As a final comment, we would like to recall some results about the break ofsymmetry of the minimizers ifΩ is the exterior of a ball centered at the origin.

The first result in this direction has been given by M. Esteban([2]) who hasshown that the global minimizer for problem I.1-I.2 withK = 1, F (u) = u2 and

G(u) = |u|p, 2 < p <2N

N − 2, is not radially symmetric. In this case, using

the fact that the minimizer is positive and with the help of a symmetrizationtechnique, M. Esteban and W. Strauss([5]) have shown that the global minimizerhas a “codimension one radial symmetry”. Roughly speaking, this means thatthe break of the radial symmetry occurs in exactly one direction in the spacevariables.

In [7] we have shown that for problem I.1-I.2, even local minimizers are notradially symmetric. As far as global minimizers are concerned, we have provedthat they enjoy the codimension one radial symmetry alluded above.

II. Statement of the results

We consider the problem of minimizing

V (u)=12

∫Ω


F (u(x)) dx(II.1)

subject to

I (u)=∫Ω

G(u(x)) dx = λ > 0(II.2)

for u in the spaceH 1K (Ω) and we make the following assumptions:

H1) Ω ⊂ IRN is an open set with bounded complement and smooth boundary∂Ω;

H2) F (u) andG(u) are real valued functions defined foru = (u1, . . . , uK ) ∈ IRK

which can be written asF (u) = Q1(u) + F1(u) andG(u) = Q0(u) + G1(u), where

Q0(u) =12〈A0u, u〉 and Q1(u) =

12〈A1u, u〉 are the quadratic parts ofF (u) and

G(u) at u = 0 (A0 andA1 are symmetricK × K matrices) andF1(u) andG1(u)areC2 functions vanishing atu = 0, as well as their first derivatives, and whosesecond derivatives are bounded byc0(|u|q−2 + |u|p−2), wherec0 is a constant and

466 O. Lopes

2 < q ≤ p <2N

N − 2(p <∞ if N = 2);

H3) the admissible setu ∈ H 1K (Ω) : I (u) = λ is not empty, that is,G(u) > 0

somewhere (see [8]);

H4) gradG(u) /= 0 for u ∈ IRK , u /= 0 and small (a manifold condition);

H5) V is bounded below on the admissible set and minimizing sequences arebounded inH 1

K (Ω).The following proposition will be useful to give a better motivation for the

last assumption that we are going to make.

II.3 Theorem. Under assumptionsH1 − H5, if u ∈ H 1K (Ω), u 6≡ 0, is the weak

limit in H 1K (Ω) of a minimizing sequence for problem II.1-II.2, thenu solves the

Euler-Lagrange system

−∆u + H ′(u) = 0,∂u∂n

= 0 on ∂Ω,

whereH (u) = F (u) +αG(u), for some real constantα. Moreover,α is such thatthe symmetric matrixA1 + αA0 is positive.

AssumptionsH1−H5 are standard for the problems we are considering; nextwe make a further assumption which has to do with the structure of the problemII.1-II.2 and will be crucial for our method.

H6) if u ∈ H 1K (Ω), u 6≡ 0, is the weak limit inH 1

K (Ω) of a minimizing sequencefor problem II.1-II.2 and

−∆u + H ′(u) = 0,∂u∂n

= 0 on ∂Ω,(II.4)

whereH (u) = F (u) + αG(u), for some real constantα, is the Euler-Lagrangeequation satisfied byu (according to the previous theorem) then there is anelementh ∈ H 1

K (Ω) (which can be taken with support contained in a ball) suchthat

W ′′(u)(h, h)=∫Ω

|gradh(x)|2dx +∫Ω

〈h(x),H ′′(u(x))h(x)〉dx < 0,(II.5)

whereW(u) = V (u) + αI (u).

II.6 Remarks.

1) ConditionH4 is satisfied if detG′′(0) /= 0 and it is a manifold condition in thesense that it guarantees that gradI (u) = G′(u(x)) 6≡ 0 if u ∈ H 1

K (Ω) andu 6≡ 0.

2) If Ω = IRN then assumptionH6 is always satisfied ([4]); but, in the case ofan exterior domain, it is not clear whether that happens or not. However, we


will show thatH6 holds for any nonlinearitiesF (u) andG(u) providedΩ is thecomplement of a ball. We will also show thatH6 holds for anyΩ provided thatthe nonlinearities are sum of homogeneous functions with appropriate signs.

3) A different way of formulating assumptionH6 is that for anyu ∈ H 1K (Ω),

u 6≡ 0, which is the weak limit of a minimizing sequence for problem II.1-II.2and satisfies II.4, the Morse index of the quadratic formW ′′(u) is at least one.

Our main result is the following:

II.7 Theorem. Under assumptionsH1 − H6, if un ∈ H 1K (Ω) is a minimizing

sequence for II.1-II.2 andun converges weakly inH 1K (Ω) to u 6≡ 0, then un

converges tou strongly in LrK (Ω), 2 < r <

2NN − 2

, and gradun converges to

gradu in L2K (Ω).

Roughly speaking, Theorem II.7 says that, under assumptionsH1 − H6, di-chotomy never occurs for problem I.1-I.2.

Now we give two sufficient conditions forH6.

II.8 Theorem. Under assumptionH2, if Ω is the exterior of a ball, then assump-tion H6 is always satisfied.

Actually, in the proof of Theorem II.8 we do not need to assume thatu isthe weak limit of a minimizing sequence. In other words, ifΩ is the exterior ofa ball andu ∈ H 1

K (Ω) is a nontrivial solution of II.4 then there is anh ∈ H 1K (Ω)

such that II.5 is satisfied. Incidentally, this shows that if we consider the vectorwave function

∂2u∂t2

−∆u + gradH (u) = 0

in the exterior of a ball with Neumann boundary condition, then any nontrivialequilibrium is linearly unstable (hence, unstable). If the equation is taken on theentire spaceRN such a result is known as Derrick’s Theorem.For the next theorem we assume thatF (u) andG(u) have the following form:

F (u) = Q1(u) +L∑

i =1

Fpi (u)

and

G(u) = Q0(u) +M∑j =1

Gqj (u)

where Q0(u) and Q1(u) are as in assumptionH2 and Fpi (u) and Gqj (u) arehomogeneous of degreespi andqj , respectively.

II.9 Theorem. In each one of the following three cases assumptionH6 is satisfied.

468 O. Lopes

i) Q0(u) is positive definite,Q1(u) is positive and has an eigenvalue equal tozero and either there is an indexi0 such that

Fpi (u) ≥ 0, Gqj (u) ≥ 0 for pi < pi0 and qj < pi0

and

Fpi (u) ≤ 0, Gqj (u) ≤ 0 for pi > pi0 and qj > pi0

or there is an indexj0 such that

Fpi (u) ≥ 0, Gqj (u) ≥ 0 for pi < qj0 and qj < qj0

and

Fpi (u) ≤ 0, Gqj (u) ≤ 0 for pi > qj0 and qj > qj0.

ii) Q0(u) is negative definite,Q1(u) ≡ 0, Fpi (u) ≥ 0 for any i and there is anindex j0 such thatpi ≤ qj0 for any i andGqj (u) ≥ 0 for qj > qj0 andGqj (u) ≤ 0for qj < qj0.

iii) Q1(u) is positive definite,Fpi (u) ≥ 0, Q0(u) ≡ 0 and eitherG(u) = Gq(u)with pi ≤ q for any i , or Gqj (u) ≥ 0 for any j andpi ≤ qj for any i and j .

II.10 Remark.In part i of the previous theorem, the condition aboutQ1(u) is justa normalization. We will come back to this point later.

For II.1-II.2 the goal is to show that minimizing sequences are precompactand, in order to accomplish that, we have to consider several cases. But first wehave to define the functional

V∞(u)=12

∫IRN

|gradu(x)|2 +∫

IRN

F (u(x))dx(II.11)

subject to

I∞(u)=∫

IRN

G(u(x))dx = λ(II.12)

for u ∈ H 1K (IRN ).

In the terminology of [1], II.11-II.12 is the problem at infinity.

First case: Q0(u) (the quadratic part ofG(u) at u = 0) is positive definite.

We start by giving a sufficient condition for assumptionH5.

II.13. Theorem. If Q0(u) is positive definite,G1(u) ≥ 0 and lim|u|→∞

F−(u)/|u|l =

0, whereF−(u) is the negative part ofF (u) and l = 2 +4N

, then assumptionH5

is satisfied.


Without loss of generality, we may assume that the quadratic partQ1(u) ofF (u) at u = 0 satisfies the followingnormalization condition :

II.14 The eigenvalues ofQ1(u) are nonnegative and at least one is zero.

In order to show this we have to notice that if in problem II.1-II.2 we replaceF (u) by F (u) − βG(u), β constant, then, due to II.2, we are simply subtractingthe constantβλ from V (u). So, replacingF (u) by F (u)−βG(u) does not changeproblem II.1-II.2. But the new quadratic part atu = 0 is Q1(u)− βQ0(u) and so,if we chooseβ as the minimum ofQ1(u) underQ0(u) = 1 then the new quadraticform meets condition II.14.

We denote byV∞(λ) the infimum ofV∞(u) defined by II.11 on the admis-sible set II.12.

II.15. Remarks(see [4])

1) Under the normalization condition II.14 we haveV∞(λ) ≤ 0 for anyλ > 0.

2) If F (u) ≥ 0 everywhere inRK thenV∞(λ) = 0 for anyλ > 0.

3) If F (u) < 0 somewhere inRK then there is aλ0 ≥ 0 such thatV∞(λ) < 0for λ > λ0.

4) If limt→0

F (tw)/|t |l = −∞, wherel = 2 +4N

andw is an eigenvector associated

to a zero eigenvalue ofQ1(u) (see II.14) thenV∞(λ) < 0 for anyλ > 0.

Now we consider a further assumption:

H7) For any solutionu ∈ H 1K (Ω) of the system

−∆u(x) + F ′(u(x)) = 0,∂u∂n

= 0 on ∂Ω(II.16)

we haveV (u) =12

∫Ω


F (u(x))dx ≥ 0.

If Ω = IRN thenH7 is always satisfied; this is a consequence of Pohozaev’sidentity. Next we give sufficient conditions forH7.

II.17 Theorem. Under assumptionH2, assumptionH7 is satisfied ifΩ is theexterior of a ball.

II.18 Theorem. Under assumptionH2,if F (u) = Q1(u) +L∑

i =1

Fpi (u), and there is

an indexi0 such thatFpi (u) ≥ 0 for pi < pi0 and Fpi (u) ≤ 0 for pi > pi0 thenassumptionH7 is satisfied.

470 O. Lopes

II.19 Remark.Notice that Theorem II.18 is consistent with Theorem II.9 parti .

Now we can state the existence result in this first case.

II.20 Theorem. Suppose thatQ0(u) is positive definite,Q1(u) satisfies the nor-malization condition II.14 andV∞(λ) < 0. Then, under assumptionsH1,H2,H5

and H6, any minimizing sequenceun for II.1-II.2 is precompact inLrK (Ω), 2 <

r <2N

N − 2, and gradun is precompact inL2

K (Ω). Moreover,un is precompact

in L2K (Ω) (and hence inH 1

K (Ω)) if either assumptionH7 is satisfied (in this casethe Lagrange multiplier is strictly positive) orV∞(µ) < 0 for anyµ > 0.

2nd case:Q0(u) is negative definite.

II.21 Theorem. BesidesH1,H2,H3 andH6, suppose thatQ0(u) is negative def-inite and eitherN ≥ 3 andF (u) ≥ 0 or N = 2, Q1(u) is positive definite andF1(u) ≥ 0. Then any minimizing sequence (un) for II.1-II.2 is precompact in

LrK (Ω), 2 < r <

2NN − 2

and gradun is precompact inL2K (Ω). Moreover un

is precompact inL2K (Ω) (and hence inH 1

K (Ω)) if either Ωc is star shaped or〈gradF (u), u〉 ≥ 0 for anyu ∈ RK .

3rd case:Q0(u) ≡ 0 andQ1(u) is positive definite.

II.22. Theorem. BesidesH1,H2,H3 and H6, suppose thatQ0(u) ≡ 0, F (u) =Q1(u)+F1(u) whereQ1(u)is positive definite,F1(u) ≥ 0 andG(u) = Gq(u)+G(u)whereGq(u) is homogeneous of degreeq > 2 and has a fixed sign (that is, eitherGq(u) > 0 or Gq(u) < 0 for u /= 0) andG(u)/Gp(u) tends to zero asu → 0.Then any minimizing sequence of II.1-II.2 is precompact inH 1

K (Ω).

II.23 Remark.If we take II.1-II.2 with Dirichlet boundary conditions (that is,uin H 1

0,K (Ω)) then the infimum is never achieved ([2]).

III. Application to the stability of solitary waves

Let H be a real Hilbert space. Aflow in H is a continuous mapϕ : IR×H → Hsuch thatϕ(0, u) = u andϕ(t , ϕ(s, u)) = ϕ(t + s, u) for any u ∈ H ands, t ∈ IR.A subsetS ⊂ H is invariant under the flow ifϕ(t ,S) = S for any realt . Aclosed, bounded and invariant setS is stable if for any ε > 0 there is aδ > 0such thatd(u,S) < δ implies d(ϕ(t , u),S) < ε for any t ∈ IR, whered(u,S)means the distance fromu to S.

A continuous functionalV : H → IR is a first integral (or a conservedquantity) for the flowϕ(·, ·) if V (ϕ(t , u)) = V (u) for any u ∈ H and t ∈ IR.


Now we make the following assumptions:

i) ϕ(·, ·) is a flow in a Hilbert spaceH that has two first integralsV (u) andQ(u)whereQ(u) is a quadratic form;

ii) if un ∈ H is a sequence such that|un| → +∞ and 0< λ1 ≤ Q(un) ≤ λ2, forsome constantsλ1 andλ2, thenV (un) → +∞;

iii) V (·) is uniformly continuous on bounded sets;

iv) for any λ > 0, any minimizing sequence ofV (·) on the setu : Q(u) = λis precompact.

Assumptions ii and iv guarantee that the minimumV (λ) of V (·) onu : Q(u) = λ is achieved and we define the (compact) setS(λ) = u ∈ H : Q(u) = λ andV (u) = V (λ).

III.1 Theorem. If a flow ϕ(·, ·) in a Hilbert spaceH has two first integralsV (u)andQ(u) satisfying assumptions i-iv then, for anyλ0 > 0, the setS(λ0) is stable.

III.2 Remark.If we know only thatQ(ϕ(t , u)) = Q(u) and V (ϕ(t , u)) ≤ V (u)for t ≥ 0 thenS(λ0) is stable for positive time.

Theorems II.20 and III.1 can be used to show the existence of a stable set ofstanding waves for certain Schrodinger equations and systems and bound statesfor Klein-Gordon equation in the exterior of a bounded domain with Neumannboundary condition (in [4] we have done such applications in the caseΩ = IRN ).

We want to remark that for the exterior domain uniqueness of the Cauchyproblem for the Schrodinger equation and Klein-Gordon equation is an openquestion. What is known is the existence of weak solutions that preserve thecharge and whose energy is a nonincreasing function oft and so, accordingwith Remark III.2, we have at least stability for positive time. However, it isa common believe that uniqueness holds in the exterior domain (at least in thesubcritical case).

The stability (and unstability) of solitary waves has been studied in [9], [10]and [11] (among many others) and, according to them, in order to prove stabilitywe have to assume the existence of a smooth familyψ(λ) of solitary waves as afunction of the chargeλ and thatd′′(λ) > 0, whered(λ) = V (ψ(λ)) +λQ(ψ(λ)).

Our alternative presentation gives the existence of a stable setS(λ0) whoseelements are solitary waves and what is left is to describe the structure of the setS(λ). For instance, ifΩ is the exterior of a ball is it true (generically speaking)that except for rotationsS(λ) is the union of a finite number of orbits?

The Schrodinger equation has been studied in [5]. The model is

i∂u∂t

+∆u + |u|p−1u = 0 for |x| > R and∂u∂n

= 0 for|x| = R,

472 O. Lopes

u(t , x) ∈ IC . The energy isV (u) =12

∫Ω

|gradu(x)|2dx − 1p + 1

∫Ω

|u(x)|p+1dx

and the charge isQ(u) =12

∫Ω

|u(x)|2dx.

If 1 < p < 1 +4N

and R is large then in [5] it is proved the existence of a

stable solitary wave.For p in the above range then Theorem II.20 applies (actually for anyΩ).Probably for homogeneous nonlinearities we can show the strict subaditivy

of V (λ) and then, for that case, we can use concentration-compactness.

IV. Proof of the results

Proof of Theorems II.3 and II.7

The proof of Theorem II.7 follows the method we have developed in [3] and [4]and it will be presented here with some modifications (improvements, we hope).We start by stating a few lemmata and from now until the end of the proof ofTheorem II.7 we assume that the assumptionsH1 − H6 are satisfied.IV.1. Lemma. Let un be a bounded sequence inH 1

K (Ω) and suppose that for

somer ∈(

2,2N

N − 2

)the sequenceun is not precompact inLr

K (Ω); then there

exist a subsequenceunj of un and a sequencecnj ∈ IRN such that|cnj | → +∞andvnj (x)=unj (x + cnj ) converges weakly inH 1

K (IRN ) to somev 6≡ 0.The proof of lemma IV.1 is a modification of the proof of well known results

due E. Lieb ([12]) and P. Lions ([1]).The proof of next lemma is elementary.

IV.2. Lemma. The functionalsV , I : H 1K (Ω) → IR are of classC2 with first

and second derivatives bounded on bounded sets and uniformly continuous onbounded sets.

Now let u(t , x), t ∈ (−δ0, δ0), δ0 > 0 be aC2 function with values inH 1K (Ω)

such that ∫Ω

G(u(t , x))dx = λ,

for t ∈ (−δ0, δ0). We will say that such a curve isadmissible. Differentiatingonce and twice with respect tot we get∫

Ω

〈gradG〈u(t , x)), u(t , x)〉dx = 0

and ∫Ω

〈u(t , x),G′′(u(t , x))u(t , x)〉dx +∫Ω

〈gradG(u(t , x)), u(t , x)〉dx = 0,


where dot means derivative with respect tot .So, if we denoteu(0, x) by u(x), we see that the elementsh andh which are

first and second derivatives, respectively, att = 0 of an admissible curve mustsatisfy ∫

Ω

〈gradG(u(x)), h(x)〉dx = 0

and ∫Ω

〈h(x),G′′(u(x))h(x)〉dx +∫Ω

〈gradG(u(x)), h(x)〉dx = 0.

We say that a pair (h, h) is admissiblefor u if h and h satisfy the twoequations above. The calculation we have performed show that ifh and h arethe first and second derivatives, respectively, of an admissible curve, then thepair (h, h) is admissible in the sense we have defined. We need the converse withsome uniformity with respect to a sequenceun.IV.3 Lemma. Let un ∈ H 1

K (Ω) be a sequence converging weakly inH 1K (Ω) to

someu 6≡ 0 and (hn, hn) be a bounded admissible sequence forun; that is, hn

and hn are bounded sequences inH 1K (Ω) satisfying:∫

Ω

〈gradG(un(x)), hn(x)〉dx = 0(IV.4)

and ∫Ω

〈hn(x), G′′(un(x))hn(x)〉dx +∫Ω

〈gradG(un(x)), hn(x)〉dx = 0.(IV.5)

Then there is aδ0 > 0 and a sequence ofC2 functions hn : (−δ0, δ0) →H 1

K (Ω) defined forn large and satisfying the following conditions: i)hn(0) =0, hn(0) = hn and hn(0) = hn; ii) for each n, un + hn(t) is an admissible curve;iii) hn(t), hn(t) and hn(t) are equicontinuous.

Proof. From assumptionH4 (see Remark II.6, number 1) we know thatgradG(u(x)) /= 0. If ψ is a smoothRK -valued function with compact supportsuch that∫Ω

〈gradG(u(x)), ψ(x)〉dx /= 0 and we consider the equation

Hn(σ, t) =∫Ω

G(un + σψ + t hn +t2

2hn)dx− λ = 0

then the implicit function theorem can be used at (0, 0) to giveσn(t), and then

hn(t) = σn(t)ψ + t hn +t2

2hn satisfies the conditions of Lemma IV.3 (the equicon-

tinuity of hn(t), hn(t) andhn(t) follows from lemma IV.2) and this proves lemmaIV.3

IV.6 Lemma. Let un ∈ H 1K (Ω) be a minimizing sequence of II.1-II.2 converging

weakly in H 1K (Ω) to someu 6≡ 0. Then

474 O. Lopes

(i) |V ′(un)| → 0 as n → +∞, the norm of the derivative being calculated onthe admissible elements (that is,the elements ofH 1

K (Ω) which are tangent to themanifold defined by the constraint atun);

(ii) if for some δ0 > 0, hn : (−δ0, δ0) → H 1K (Ω) is a sequence ofC2 curves

such thathn(0) = 0, un + hn(t) is admissible,hn(0) and hn(0) are bounded andhn(t), hn(t) and hn(t) are equicontinuous then

lim infd2

dt2V (un + hn(t))t=0 ≥ 0.(IV.7)

Lemma IV.6 has been also used in [13] and its proof is a sort of elementarycalculus argument.

Now let u ∈ H 1K (Ω) be a generic element such thatI (u) = λ. In order to

compute|V ′(u)| in the admissible directions we have to maximize

V ′(u)ϕ =∫Ω

〈gradu(x), gradϕ(x)〉dx +∫Ω

〈gradF (u(x)), ϕ(x)〉dx

for ϕ ∈ H 1K (Ω) subject to∫

Ω

〈gradG(u(x)), ϕ(x)〉dx = 0

and ∫Ω

(|gradϕ(x)|2 + |ϕ(x)|2)dx = 1.

If ϕ is the place where the maximum is achieved, then there are real numbersα andγ such that∫

Ω

(〈gradu(x), gradϕ(x)〉 + 〈gradF (u(x)) + αgradG(u(x)), ϕ(x)〉

(IV.8)

+γ〈gradϕ(x), gradϕ(x)〉 + γ〈ϕ(x), ϕ(x)〉)dx = 0

for anyϕ ∈ H 1K (Ω). In particular,

−∆(u + γϕ) + gradF (u(x)) + αgradG(u(x)) + γϕ = 0(IV.9)

and

∂

∂n(u + γϕ) = 0 on ∂Ω.(IV.10)

If we setϕ = ϕ in IV.8 we getV ′(u)ϕ = −γ and this shows that|V ′(u)| = −γ.Moreover, if h(t) is a smooth curve withh(0) = 0 such thatu + h(t) is

admissible then


d2

dt2V (u + h(t))t=0 =

∫Ω

|gradh(x)|2dx(IV.11)

+∫Ω

(〈h(x), (F ′′(u(x)) + αG′′(u(x)))h(x)〉 − γ〈gradϕ(x), gradh(x)〉−γ〈ϕ(x), h(x)〉)dx,

whereh and h are the first and second derivatives, respectively, ofh(t) at t = 0.

IV.12 Lemma. Let L0 : D(L0) ⊂ L2K (Ω) → L2

K (Ω) be defined byL0 = −∆ + B0,where B0 is a K × K constant symmetric matrix with eigenvaluesµ1, . . . , µK

and D(L0) = u ∈ H 2K (Ω) :

∂u∂n

= 0 on ∂Ω. Then the spectrumσ(L0) of L0 is

[ min1≤i≤K

µi ,+∞).

Proof. If M denotes an orthogonal matrix that diagonalizesB0, we see that theequation−∆u + B0u−µu = f is equivalent to−∆v + D0v−µv = g through thechange of variableu = M v, whereD0 = diag(µi ) and then lemma IV.12 followsfrom the fact that the spectrum of the scalar−∆ operator with domain

u ∈ H 2(Ω) :∂u∂n

= 0 on∂Ω is [0,+∞) (becauseΩc is bounded).

IV.13 Lemma. Let L : D(L) ⊂ L2K (Ω) → L2

K (Ω) be defined byL = −∆ + B0 +B(x) = L0 + B(x), whereL0 is as in the previous lemma,B(x) is a continuousK × K symmetric matrix that tends to zero as|x| → +∞ and D(L) = D(L0).Suppose there is a vectorz ∈ L2

K (Ω) such that〈Lh, h〉 ≥ 0 for any h ∈ D(L)such that〈h, z〉 = 0. Theni) the spectrumσ(L) of L cannot have two distinct strictly negative elements;ii) the eigenvalues ofB0 are all nonnegative.

Proof. Part i is very well known ([3], [14]). Part ii follows from part i, the pre-vious lemma and from the fact thatL and L0 have the same essential spectrum(see [15], Theorem 5.7 page 304).

Let un and u be as in theorems II.3 and II.7. From lemma IV.6, equationsIV.8, IV.9 and IV.10 for u = un we know that there are sequencesαn, γn andϕn such thatγn → 0, |ϕn|H 1

K (Ω) = 1 and∫Ω

(〈gradun(x), gradϕ(x)〉 + 〈gradF (un(x)) + αngradG(un(x)), ϕ(x)〉

(IV.14)

+γn〈gradϕn(x), gradϕ(x)〉 + γn〈ϕn(x), ϕ(x)〉)dx = 0

for anyϕ ∈ H 1K (Ω),

476 O. Lopes

−∆(un(x) + γnϕn(x)) + gradF (un(x)) + αngradG(un(x))(IV.15)

+γnϕn(x) = 0

and

∂

∂n(un(x) + γnϕn(x)) = 0 on ∂Ω.(IV.16)

IV.17 Lemma. The sequenceαn is bounded.Proof. If |αn| → +∞ for some subsequence (for which we keep the same nota-tion) and we divide IV.14 byαn and letn tend to infinity we get∫

Ω

〈gradG(u(x)), ϕ(x)〉dx = 0 for any ϕ ∈ H 1K (Ω)

and this is a contradiction with assumptionH4 (see Remark II.6, number 1) andthis proves the lemma.IV.18 Lemmai) If hn is a bounded sequence of elements ofH 1

K (Ω) such that∫Ω

〈gradG(un(x)), hn(x)〉 dx = 0 then

lim inf (∫Ω

|gradhn|2 dx +∫Ω

〈hn(x), (F ′′(un(x)) + αnG′′(un(x)))hn(x)〉 dx) ≥ 0.

ii) if h ∈ H 1K (Ω),

∫Ω

〈gradG(u(x)), h(x)〉 dx = 0 andαn → α then∫Ω

|gradh|2 dx +∫Ω

〈h(x), (F ′′(u(x)) + αG′′(u(x))h(x)〉 dx) ≥ 0.

Proof. Let ψ be a smoothRK valued function with compact support such that∫Ω

〈gradG(u(x)), ψ(x)〉 dx /= 0 and let us define a sequence of real numbersdn

such that∫Ω

(〈hn(x),G′′(un(x))〉+ dn〈gradG(un(x)), ψ(x)〉) dx = 0. The sequence

dn is well defined forn large and then the pair (hn, dnψ) is admissible in thesense of lemma IV.3. Ifhn(t) is the sequence given by that lemma then part ifollows from IV.7 and IV.11 ( foru = un andh(t) = hn(t)) becauseγn → 0.

In order to show part ii we have to notice that if∫Ω

〈gradG(u(x)), h(x)〉 dx = 0

and we defineεn in such a way thathn = h + εnψ satisfies∫Ω

〈gradG(un(x)), hn(x)〉 dx = 0, then part ii follows from part i becauseεn → 0.

Proof of Theorem II.3.From lemma IV.17 we know thatαn is bounded and so,passing to a subsequence if necessary, we may assume thatαn converges to someα. If we let n →∞ in IV.14 we get

Minimization problems for the exterior domain 477∫Ω

(〈gradu(x), gradϕ(x)〉 + 〈(gradF (u(x)) + αgradG(u(x))), ϕ(x)〉) dx = 0

for anyϕ ∈ H 1K (Ω) and this proves thatu satisfies

−∆u(x) + H ′(u(x)) = 0,∂u∂n

= 0 on∂Ω,(IV.19)

whereH (u) = F (u) + αG(u) and this shows the first part of Theorem II.3.Now, due to the regularity results for elliptic systems we know thatu ∈

W3,sK (Ω) for 2 ≤ s < ∞. In particular,u(x) is continuous and tends to zero as

|x| tends to∞ and thenH ′′(u(x)) = A1 + αA0 + B(x), whereB(x) is continuousand tends to zero as|x| tends to∞ and so, the final statement in Theorem II.3follows from lemmata IV.13 and IV.18, part ii.

Proof of Theorem II.7.In order to show the precompactness ofun in LrK (Ω), r ∈

(2,2N

N − 2), we argue by contradiction. Suppose that for somer ∈

(2,

2NN − 2

)the sequenceun is not precompact inLK

r (Ω). From lemma IV.1, the bounded-ness ofαn and passing to a subsequence if necessary, we may assume thatαn

converges toα and thatvn(x)=un(x + cn) converges weakly inH 1K (IRN ) to some

v /= 0, for some sequencecn ∈ RN such that|cn| → +∞.If we pass to the limit asn → +∞ in IV.14 we get

−∆u(x) + H ′(u(x)) = 0,∂u∂n

= 0 on ∂Ω(IV.20)

whereH (u) = F (u) + αG(u). If in IV.14 we make the change of variablex →x + cn and pass to the limit asn → +∞ we get

−∆v(x) + H ′(v(x)) = 0.(IV.21)

From assumptionH6 we know that there is an elementh which can be takento be smooth and with support in a ball such that

W ′′(u)(h, h) =∫Ω


〈h(x),H ′′(u(x))h(x)〉dx < 0.(IV.22)

Moreover, sincev 6≡ 0 satisfies IV.21 andN ≥ 2 we also know ([4]) thatthere is an elementk which can be taken smooth with compact support such that

W′′∞(v)(k, k) =

∫IRN

|gradk(x)|2dx +∫

IRN

〈k(x),H ′′(v(x))k(x)〉dx < 0(IV.23)

Next we definehn(x) = anh(x) + bnk(x − cn) and we choosean and bn in

such way thata2n + b2

n = 1 and∫Ω

〈gradG(un(x)), hn(x)〉dx = 0. For n largeh(.)

andk(.− cn) have disjoint supports and so, forn large we have

478 O. Lopes

∫Ω

|gradhn(x)|2dx +∫Ω

〈hn(x), (F ′′(un(x)) + αnG′′(un(x)))hn(x)〉dx =

a2n

(∫Ω


〈h(x), (F ′′(un(x)) + αnG′′(un(x)))h(x)〉dx

)+b2

n

(∫IRN

|gradk(x)|2dx +∫

IRN

〈k(x), (F ′′(vn(x)) + αnG′′(vn(x)))k(x)〉dx

)(IV.24)

To obtain this last integral, the change of variablesx → x − cn has beenperformed.

Sinceun u in H 1K (Ω), vn v in H 1

K (IRN ), αn → α andh(x) andk(x)are fixed elements with compact support we have

limn→+∞

(∫Ω


〈h(x), (F ′′(un(x)) + αnG′′(un(x)))h(x)〉dx

)= W′′(u)(h, h) < 0

and

limn→+∞

(∫IRN

|gradk(x)|2dx +∫

IRN

〈k(x), (F ′′(vn(x)) + αnG′′(vn(x)))k(x)〉dx

)= W ′′

∞(v)(k, k) < 0

and then, from IV.24,

lim inf(∫Ω

|gradhn(x)|2 +∫Ω

〈hn(x),F ′′((un(x)) + αnG(un(x)))hn(x)〉) dx < 0,

a contradiction in view of lemma IV.18, part i. This shows the precompactness

of un in LrK (Ω), 2 < r <

2NN − 2

, and hence, the convergence ofun to u in

LrK (Ω), 2< r <

2NN − 2

.

Next we show that gradun to gradu in L2K (Ω). If we subtract IV.15 from

IV.20, take the scalar product withu − un − γnϕn, integrate inΩ and use theboundary conditions forun + γnϕn andu we get:∫

Ω

|grad(un(x)− u(x))|2dx +∫Ω

〈(un(x)− u(x)),B0(un(x)− u(x))〉dx

= −γn

∫Ω

〈gradϕn, grad(un − u)〉dx

−∫Ω

〈gradF1(un(x))− gradF1(u(x)), un(x)− u(x)〉dx

+αn

∫Ω

〈gradG1(un(x))− gradG1(u(x)), un(x)− u(x)〉dx +

(αn − α)∫Ω

〈gradG(u(x)), un(x)− u(x)〉dx− γn

∫Ω

〈ϕn(x), (un(x)− u(x))〉dx


whereB0 = A1 + αA0, and the right hand side goes to zero asn → +∞ because

αn → α, γn → 0, un → u in LrK (Ω) for 2 < r <

2NN − 2

and assumptionH2.

This shows that gradun → gradu in L2K (Ω) becauseB0 is positive (lemma IV.13,

part ii) and Theorem II.7 is proved.

IV.25 Remark.If we know in advance thatB0 is actually positive definite, thenthe argument above also shows thatun → u in L2

K (Ω).

Proof of Theorem II.8.For simplicity we assume that the ball is centered at theorigin.

First we prove a Pohozaev’s identity for solutions of II.4. As we have pointedout, due to growth assumptionsH2, u belongs toW3,s

K (Ω), for 2 ≤ s < ∞.

Formally, it we take the scalar product of II.4 withN∑

i =1

xi∂u∂xi

and integrate inΩ

we get the Pohozaev’s identity

(N − 2

2)∫Ω

|gradu(x)|2dx + N∫Ω

H (u(x))dx

(IV.26)

−∫∂Ω

(12|gradu(x)|2 + H (u(x)))(

N∑i =1

xi ni )dS = 0

wheren(x) = (n1(x), . . . , nN (x)) is the outer normal toΩ. Although IV.26 makes

sense, since we do not know whetherN∑

i =1

xi∂u∂xi

is integrable onΩ, this procedure

has to be justified. This can be done by introducing a convenient sequence ofcut-off functionsϕn(x) (see [16], chapter 6); so, if we take the scalar product of

II.4 with ϕn(x)

(N∑

i =1

xi∂u∂xi

), integrate inΩ and pass to the limit asn goes to

infinity we get IV.26.

Next we defineh =N∑

i =1

xi∂u∂xi

and we, formally, compute

W ′′(u)(h, h) =∫Ω

∣∣∣∣∣gradN∑

i =1

xi∂u∂xi

∣∣∣∣∣2

dx +

∫Ω

⟨(N∑

i =1

xi∂u∂xi

),H ′′(u(x))

(N∑

i =1

xi∂u∂xi

)⟩dx.

If we use II.4 and IV.26 then after a few integration by parts we get

480 O. Lopes

W ′′(u)(h, h) = (2− N )∫Ω

|gradu(x)|2dx + 2∫∂Ω

|gradu(x)|2(

N∑i =1

xi ni

)dS

+∫∂Ω

N∑i ,j =1

xi xj 〈gradu(x), grad∂u∂xj

〉ni dS

+∫∂Ω

N∑i ,j =1

xi xj 〈 ∂u∂xj

, gradH (u(x))〉ni dS(IV.27)

Now supposeΩ is the complement of the ball centered at the origin with

radiusR; in this case we haveni = −xi

Rand then

N∑j =1

xj∂u∂xj

= −R∂u∂n

= 0 on

∂Ω and so IV.27 becomes

W ′′(u)(h, h) = (2− N )∫Ω

|gradu(x)|2dx− 2R

∫∂Ω

|gradu(x)|2dS

(IV.28)

−∫∂Ω

N∑j =1

nej 〈gradu(x), grad

∂u(x)∂xj

〉 dS

where (nej ) is theexterior normalto the ball.

According to [17], page 269, equation (12), ifv(x) is a real valued function

such that∂v

∂n(x) = 0 on ∂Ω, then at any pointx ∈ ∂Ω (which can be assumed

to be the origin of the coordinate system) we have

N∑j =1

nej 〈gradv(x), grad

∂v

∂xj(x)〉 = 〈gradv(x),

∂

∂ngradv(x)〉 =

= −N−1∑i =1

N−1∑j =1

∂2g(0)∂xi ∂xj

∂v(0)∂xi

∂v(0)∂xj

whereg(x1, x2, . . . , xN1) is a function such thatg(0, 0, . . . , 0) = 0, xN = g(x1, x2,. . . , xN−1) describes the boundary and the−xN axis is the exterior normal at(0, 0, . . . , 0). In our case

g(x1, . . . , xN−1) = R−√

R2 − x21 − · · · − x2

N−1 and then∂2g(0)∂xi ∂xj

=δij

R

and then IV.28 becomes

W ′′(u)(h, h) = (2− N )∫Ω

|gradu(x)|2dx− 1R

∫∂Ω

|gradu(x)|2dS(IV.29)


SupposeN = 2 and gradu(x) = 0 on ∂Ω. If we use polar coordinates, we

have∂u∂θ

(R, θ) = 0 and∂u∂r

(R, θ) = 0 and then∂

∂r∂u∂θ

(R, θ) = 0. So, if we define

w =∂u∂θ

, an easy computation shows thatw satisfies the linear equation

−∆w+H ′′(u(x))w = 0 and sincew =∂w

∂n= 0 on∂Ω, by the unique continuation

principle,w = 0 onΩ and thenu(x) is a radial function.So, if we assume thatN ≥ 3 or N = 2 andu(x) is not a radial function, then

by IV.29 W′′(u)(h, h) < 0 and this shows that II.5 is satisfied by the functionhwe have defined.

However, in general, because of the factorxi , we do not know whetherhbelongs to the spaceH 1

K (Ω) and so, the elementh we have defined cannot beused to test II.5.

In order to justify the procedure we introduce, as before, a convenient se-quence of cut-off functionsϕn(x) and we definehn(x) = ϕn(x)h(x). Then anelementary but tedious calculation shows that

limn→+∞W ′′(u)(hn, hn) = (2− N )

∫Ω

|gradu(x)|2dx− 1R

∫∂Ω

|gradu(x)|2dS< 0

and thenW ′′(u)(hn, hn) < 0 if n is large and, sincehn ∈ H 1K (Ω) for n finite

(becauseh ∈ H 2K (Ω) andϕn is compactly supported), we see that II.5 is satisfied

with h = hn, n large but fixed. This proves Theorem II.8 in the case above.All is left is to consider the caseN = 2 andu(.) a radial function (actually

we can show that this case cannot occur but it is simpler to handle it).So supposeu(r ) is a radial function andN = 2. Then II.4 becomes

− u′′(r )− 1r

u′(r ) + gradH (u(r )) = 0, u′(R) = 0.(IV.30)

If we seth = ur and differentiate IV.30 with respect tor we get

− h′′(r )− 1r

h′(r ) + H ′′(u(r ))h(r ) = −h(r )r 2

(IV.31)

If we take the scalar product of IV.31 withrh(r ), integrate fromR to +∞and use the boundary conditionh(R) = 0 we find

W ′′(u)(h, h) = −∫ +∞

R

u2r (r )r

dr < 0

and then, by a density argument, Theorem II.8 is proved in all cases.

Proof of Theorem II.9Now suppose thatu(x) 6≡ 0 is the weak limit inH 1K (Ω)

of a minimizing sequence and satisfies II.4. Suppose also that there is a constantβ > 1 such that

〈u,H ′′(u)u〉 ≤ β〈u, gradH (u)〉(IV.32)

482 O. Lopes

for any u ∈ IRK . If we take the scalar product of II.4 withu, integrate inΩ, usethe boundary condition and IV.32 we get

W ′′(u)(u, u) =∫Ω


〈u(x),H ′′(u(x))u(x)〉dx

≤∫Ω

|gradu(x)|2dx + β∫Ω

〈u(x), gradH (u(x))〉dx

= (1− β)∫Ω

|gradu(x)|2dx < 0

and this shows that II.5 is satisfied withh = u.So, in order to prove Theorem II.9 we have to show that, under the assump-

tions we have made, condition IV.32 is satisfied .Let us recall that a functionFp(u) to be homogeneous of degreep means that

Fp(λu) = λpFp(u) for λ > 0. If we differentiate once and twice this equality withrespect toλ and we setλ = 1 we get〈u, gradFp(u)〉 = pFp(u) and〈u,F ′′p (u)u〉 =p(p − 1)Fp(u).

In order to prove Theorem II.9 it is sufficient to show that there is a constantβ > 1 such that

2Q1(u) +L∑

i =1

pi (pi − 1)Fpi (u) + α[2Q0(u) +K∑

j =1

qj (qj − 1)Gqj (u)] ≤

β2Q1(u) +L∑

i =1

pi Fpi (u) + α[2Q0(u) +K∑

j =1

qj Gqj (u)].(IV.33)

In part i we must haveα ≥ 0 (because the matrixA1 +αA0 has to be positiveaccording to Theorem II.3). So, under the assumptions of part i, IV.33 is satisfiedwith β = pi0 − 1 in the first case andβ = qj0 − 1 in the second.

Similarly, in part ii, we haveα ≤ 0 and then IV.33 is a satisfied withβ = qj0 − 1.

In part iii, if G(u) = Gq(u) has a single homogeneous term then IV.33 issatisfied withβ = q− 1. In the second case, if we take the scalar product of II.4with u and integrate we getα ≤ 0 and then IV.33 is satisfied if we chooseβ insuch way thatpi −1≤ β ≤ qj −1, for anyi andj , and this proves Theorem II.9.

Proof of Theorem II.13.SinceA0 is positive definite andG1(u) ≥ 0, the con-straint gives a bound for theL2

K (Ω) norm of u and then Theorem II.13 followsfrom interpolation inequalities (see [1],part II, page 226).

Proof of Theorem II.17.SupposeΩ is the exterior of a ball centered at the originand with radiusR. In this case, Pohozaev’s identity IV.26 withα = 0 becomes


N − 22

∫Ω

|gradu(x)|2 dx + N∫Ω

F (u(x)) dx

+R∫∂Ω

(12|gradu(x)|2 + F (u(x))) dS = 0

and then, solving for∫Ω

F (u(x)) dx , we get

V (u) =12

∫Ω

|gradu(x)|2 dx +∫Ω

F (u(x)) dx

=1N

∫Ω

|gradu(x)|2 dx− RN

∫∂Ω

(12|gradu(x)|2 + F (u(x))) dS

In order to express the integral on the boundary ofΩ in terms of an integral

onΩ, we take the scalar product of II.4 withr−NN∑

i =1

xi∂u∂xi

, r =√

x21 + · · · + x2

N ,

integrate inΩ and we get

RN

∫∂Ω

(12|gradu(x)|2 + F (u(x))) dS = −RN

∫Ω

r−N−2(N∑

i =1

xi∂u∂xi

)2 dx

+RN

N

∫Ω

r−N |gradu(x)|2 dx.

If we replace this expression in IV.34 we find

V (u) =1N

∫Ω

(1− RN

r N)|gradu(x)|2 dx + RN

∫Ω

r−N−2(N∑

i =1

xi∂u∂xi

)2 dx

and thenV (u) > 0 becauser = |x| > R for x ∈ Ω and this proves TheoremII.17.

Proof of Theorem II.18.SupposeF (u) satisfies the following condition:

F (u) ≥ γ〈gradF (u), u〉(IV.34)

for any u ∈ RK and someγ < 1/2.If we take the scalar product of II.16with u, integrate inΩ, use the boundary condition foru and IV.35 we get

V (u) > (1/2− γ)∫Ω

|gradu(x)|2 dx > 0. So, in order to prove the theorem, it is

sufficient to show that, under the assumptions we have made, IV. 35 is satisfied.But IV.35 is equivalent to

Q1(u) +l∑

i =1

Fpi ≥ γ(2Q1(u) +L∑

i =1

pi Fpi (u))

and it is easy to see that this inequality is satisfied withγ =1

pi0, and this

proves Theorem II.18.

484 O. Lopes

Proof of Theorem II.20.Next lemma will be usefull and its proof can be foundin [2].

IV.36 Lemma. Under assumptionsH1−H5, any weak limit inH 1K (Ω) of a mini-

mizing sequence for II.1-II.2 is not identically zero if and only ifV (λ) < V∞(λ).

IV.37 Lemma. Under the assumptions of Theorem II.20 we haveV (λ) < V∞(λ).Proof. We follow the method present in [2]. Under the assumptions of The-orem II.20 we know thatV∞(λ) is achieved at some elementu0(r ) which isradially symmetric (except for translation) (see [4]); moreover,u0(x) satisfies theEuler system−∆u(x) + gradF (u(x)) + βgradG(u(x)) = 0, with β > 0; in partic-ular, u0(r ) /= 0 if r is large because (u0(r ), u′0(r )) belongs to the stable manifoldof (0, 0).

Let cn ∈ IRN be a sequence such that|cn| → +∞ and we try to solve∫Ω

G(u0(xσ

+cn))dx = λ with σ close to 1; this equation is equivalent toσN −1 =

1λ

∫Ωc

G(u0(xσ

+ cn))dx. If we definegn(σ) =1λ

∫Ωc

G(u0(xσ

+ cn))dx we see that

gn(σ) > 0 for n large becauseG(u0(xσ

+ cn)) = Q0(u0(xσ

+ cn)) + G1(u0(xσ

+

cn)),Q0(u) is positive definite and limu→0

G1(u)Q0(u)

= 0. Sincegn(σ) tends to zero as

n → +∞ uniformly for, say,12< σ < 2 and

ddσ

(σN −1)

∣∣∣∣σ=1

= N /= 0, then there

exists a sequenceσn > 1 such thatσn → 1 andσNn −1 =

1λ

∫Ωc

G(u0(xσn

+cn))dx,

and this means thatvn(x) = u0(xσ n

+ cn) is admissible for II.1-II.2.

Next we claim that

V (vn) =12

∫Ω

|gradu0(xσn

+ cn)|2dx +∫Ω

F (u0(xσn

+ cn))dx <

(IV.38)12

∫IRN

|gradu0(x)|2dx +∫

IRN

F (u0(x))dx = V∞(λ)

for n large. In fact, IV.38 is equivalent to

−12

∫Ωc

|gradu0(xσn

+ cn)|2dx−∫Ωc

F (u0(xσn

+ cn))dx <(IV.39)

12

(1− σN−2n )

∫IRN

|gradu0(x)|2dx + (1− σNn )∫

IRN

F (u0(x))dx

If we divide this inequality by 0< σNn − 1 =

1λ

∫Ωc

G(u0(xσn

+ cn))dx and

pass to the limit asn → +∞, the right hand side converges to

− (N − 2)2N

∫IRN

|gradu0(x)|2dx−∫

IRN

F (u0(x))dx = βλ > 0


(we have used Pohozaev’s identity); moreover, since

lim sup− ∫

Ωc F (u0( xσn

+ cn))dx∫Ωc G(u0( x

σn+ cn))dx

≤ 0 (becauseQ1(u) ≥ 0 due to the normalization

condition II.14), we see that IV.39 (hence IV.38) is satisfied forn large, and thisproves the claim and the lemma.

Proof of Theorem II.20.Let un be a minimizing sequence for II.1-II.2. FromassumptionH5 and passing to a subesequence if necessary, we may assume thatun converges weakly inH 1

K (Ω) to someu. From lemmata IV.36 and IV.37 weconclude thatu 6≡ 0 and then Theorem II.7 applies. In particularV (u) ≤ V (λ) <

0 because the functional12

∫Ω


Q1(u(x))dx is convex due the

normalization condition II.14.

We also know thatu satifies the Euler system IV.20 and thatα ≥ 0 becausethe matrixB0 = A1 + αA0 is positive.

If H7 is satisfied thenα has to be strictly positive (becauseV (u) < 0) and thenB0 is positive definite and the convergence inL2

K (Ω) follows from Remark IV.25.

Now supposeV∞(µ) < 0 for any µ > 0 and that∫Ω

Q0(u(x))dx <

lim∫Ω

Q0(un(x))dx (se can pass to a subsequence if necessaty). ThenI (u) =

λ0 < λ. From concentration-compactness ([1]) we know thatV (λ) ≤ V (λ0) +V∞(λ − λ0) and thenV (λ0) < V (u) ≤ V (λ) ≤ V (λ0) + V∞(λ − λ0), a contra-diction and Theorem II.20 is proved.

Proof of Theorem II.21.First we notice thatH5 is satisfied. IfN = 2 that isclear. If N ≥ 3 andV (u) is bounded then|gradun|L2

K (Ω) is also bounded and thisgives a bound for|u|L2∗

K(Ω) and then the constraint together with an interpolation

inequality gives a bound for|u|L2K (Ω) ( see [8],part I, page 324-325).

Next we claim thatV (λ) < V∞(λ). In fact we know thatV∞(λ) is achieved atan elementu0 ∈ H 1

K (IRN ) (which can be taken radially symmetric) that satisfiesthe Euler system−∆u0(x) + gradF (u0(x)) + βgradG(u0(x)) = 0 with β ≤ 0.However, Pohozaev’s identity implies thatβ cannot be 0 and thenβ < 0, u0(r )tends to zero exponentially asr → +∞ and u0(r ) /= 0 if r is large because(u0(r ), u′0(r )) belongs to the stable manifold of (0,0).

Now in order to show thatV (λ) < V∞(λ), we argue exactly as in the proofof Theorem II.20 and we define a sequenceσn tending to 1 asn → +∞ such

that σNn − 1 =

1λ

∫Ωc

G(u0(xσn

+ cn))dx < 0 so thatu0(xσn

+ cn) is admissible

and we have to verify IV.39 forn large. If we divide IV.39 by 0< 1− σNn =

−1λ

∫Ωc

G(u0(xσn

+ cn))dx, we have to show that

486 O. Lopes

λ∫Ωc |gradu0( x

σn+ cn)|2dx

2∫Ωc G(u0( x

σn+ cn))dx

+λ∫Ωc F (u0( x

σn+ cn))dx∫

Ωc G(u0( xσn

+ cn))dx

<(1− σN−2

n )2(1− σN

n )

∫IRN

|gradu0(x)|2dx +∫

IRN

F (u0(x))dx

for n large. But, clearly, the lim sup of the left hand side is≤ 0 and the limit ofthe right hand side is

N − 22N

∫IRN

|gradu0(x)|2 +∫

IRN

F (u0(x))dx = −βλ > 0

and this shows thatV (λ) < V∞(λ).So, the first part of Theorem II.21 follows from lemma IV.36 and Theo-

rem II.7.In order to show the convergence inL2

K (Ω) we have to notice that ifu is theweak limit of a minimizing sequence thenu satisfies the Euler system

−∆u + +gradF (u(x)) + αgradG(u(x)) = 0(IV.40)

with α ≤ 0 (becauseA1 +αA0 is positive,A1 = 0 andA0 is negative definite and,according to Remark IV.25, the convergence takes place inH 1

K (Ω) if α < 0. Soall we have to show is thatα /= 0. If Ωc is star shaped (say, we respect to theorigin) this follows from Pohozaev’s identity IV.26 (because〈x, n〉 ≤ 0) and if〈gradF (u), u〉 ≥ 0 this follows from taking the scalar product ofu(x) with theequation IV.40 and integrating inΩ. This proves Theorem II.21.

Proof of Theorem II.22.Clearly H5 is satisfied. Next we show thatV (λ) <V∞(λ). We consider only the caseGp(u) > 0 for u /= 0 (the other is similar). Weknow thatV∞(λ) is achieved at an elementu0(x) ∈ H 1

K (IRN ) which is radiallysymmetric, satisfies the Euler system−∆u0(x) + gradF (u0(x)) +βgradG(u0(x)) =0, decays to zero exponentially asr → +∞ and does not vanish forr large (see

[4]). We define a sequenceσn such thatσn → 1 andσNn = 1 +

1λ

∫Ωc

G(u0(xσn

+

cn))dx. We haveσNn − 1 =

1λ

∫Ωc

G(u0(xσn

+ cn))dx > 0 and so, if we divide

IV.39 by σNn − 1 we have to show that

−λ

2

∫Ωc |gradu0( x

σn+ cn))dx∫

Ωc G(u0( xσn

+ cn))dx− λ

∫Ωc F (u0( x

σn+ cn))dx∫

Ωc G(u0( xσn

+ cn))dx

<(1− σN−2

n )2(σN

n − 1)

∫IRN

|gradu0(x)|2dx−∫

IRN

F (u0(x))dx

Since the left hand side tends to−∞ as n → +∞ (becauseF (u0(xσn

+

cn))/G(u0(xσn

+ cn)) tends to +∞) and the right hand side tends toβλ, the

inequality above is satisfied forn large and this showsV (λ) < V∞(λ).The conclusion follows from Theorem II.7 and Remark IV.25, and Theorem

22 is proved.


Proof of Theorem III.1.ClearlyS(λ0) is compact and invariant. By contradiction,suppose there areε0 > 0 and sequencesun and tn such thatd(un,S(λ0)) tendsto zero andd(ϕ(tn, un),S(λ0)) ≥ ε0. We define the sequencesσn > 0 such that

σ2n =

λ0

Q(un)=

λ0

Q(ϕ(tn, un))andvn = σnϕ(tn, un). ThenQ(vn) = λ0 andV (vn) →

V (λ0) becauseσn → 1, V (ϕ(tn, un)) = V (un) → V (λ0) andϕ(tn, un) is bounded.We conclude thatd(vn,S(λ0)) tends to zero and, hence,d(ϕ(tn, un),S(λ0)) tendsto zero, a contradiction. This proves Theorem III.1.

References

1. P.L. Lions, The Concentration-Compactness Principle in the Calculus of Variations, Ann. Inst.H. Poincare, Anal. non Lineaire, sec A,1(1984), Part I 109–145, Part II 223–283

2. M. Esteban, Nonsymmetric ground states of symmetric variational problems, Comm. Pure Appl.Math. XLIV , No. 2 (1991), 259–274

3. O. Lopes, A Constrained Minimization Problem with Integrals on the Entire Space, Bol. Soc.Bras. Mat.25, N. 1, 77–92

4. O. Lopes, Variational Systems Defined by Improper Integrals, preprint5. M. Esteban, W. Strauss, Nonlinear bound state outside an insulated sphere, Commun. in Partial

Differential Equations19 (182) (1994), 177–1976. T. Cazenave, P.L. Lions, Orbital Stability of Standing waves for Some Nonlinear Schrodinger

Equations, Comm. Math. Phys.85 (1982), 549–5617. O. Lopes, Radial, Nonradial for Some Radially Symmetric Functionals, Eletronic J. of Differ-

ential Equations (http://ejde.math.swt.edu), 1996, No. 3, 1–148. H. Beresticky, P.L. Lions, Nonlinear Scalar Fields Equations, I and II, Arch. Rat. Mech. Anal.

82 (1983), 313–3769. M. Grillakis, J. Shatah, W. Strauss, Stability of Solitary Waves in the Presence of Symmetry I,

J. Functional Analysis74 (1987), 160–19710. M. Weinstein, Liapunov Stability of Ground States of Nonlinear Dispersive Evolution Equations,

Commun. Pure Appl. Math39 (1986), 51–6811. V. Makhankov, Dynamics of classical Solitons, Phys. Reports35 (1978), 1–12812. H. Brezis, E. Lieb, Minimum Action Solutions of Some Vector Field Equations, Comm. Math.

Phys.96 No. 1 (1984), 97–11313. P.L. Lions, Solutions of Hartree-Fock Equations for Coulomb Systems, Commun. Math. Phys.

109 (1987), 33–9714. S. Coleman, V. Glazer, A. Martin, Action Minima among to a class of Euclidean Scalar Field

Equations, Comm. Math. Phys.58 (2) (1978), 211–22115. T. Kato, Perturbation Theory for Linear Operators, Springer-Verlag, New York, 196616. O. Kavian, Introductiona la Theorie des Points Critiques, Mathematiques et Applications

(SMAI), Springer-Verlag, 199317. R. Casten, C. Holland, Instability results for a reaction-diffusion equation with Neumann bound-

ary condition, J. Diff. Eq.27 (1978), 266–273

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Minimization problems for the exterior domain