Sign-Changing Solutions 9 Sign-Changing Solutions In this chapter, we study the existence of sign-changing solutions for nonlinear elliptic equations via linking methods. A linking

Chapter 9

Sign-Changing Solutions

In this chapter, we study the existence of sign-changing solutions for nonlinear elliptic equations via linking methods. A linking type theorem is established with the location of the critical point in terms of the cone structure of the space. The abstract theorem is applied to elliptic equations that have jumping nonlinearities. Under stronger conditions, we show that the existence of sign-changing critical points can be independent of the Fucik spectrum.

9.1 Linking and Sign-Changing Solutions

Let ^ be a Hilbert space and let X C ^ be a Banach space densely embedded in E. Assume that E has a closed convex cone PE and that P := PE H X has

o

interior points in X, i.e., P =P UdP in X. We use || • || and || • ||x to denote the norms in E and X respectively. We also use dist£;(-, •) and distx(-, •) to denote the distances in E and X respectively.

Let ^ G C'^{E, R). For a,b,ce R, denote

K: = {xeE: ^\x) = 0};

^^ = {xeE: ^{x) < b};

)Cc = {x e K : ^{x) = c};

X:([a,6]) = {xeK:: ^{x) G [a, 6]}.

Since || 11^1? the following negative gradient flow a for ^ is well

deflned for (t, i ) G R x ^ :

196 CHAPTER 9. SIGN-CHANGING SOLUTIONS

Suppose (Ai) JCcX and V^ : X ^ X is in CK

Under this assumption, a{t, x) is continuous in (t, x) G R x X and a{t, x) G X for X G X. With the flow a, we cafl a subset A C E wci invariant set if cr{t,A) C A for t > 0.

Definition 9.1. Let V C X be an invariant set under a. V is said to he an admissible invariant set for ^ if

o

(a) V is the closure of an open set in X, i.e., V =V UdV;

(h) if Un = cr(t^, v) ^ u in E as tn ^ oo for some v ^ V and u e JC, then Un ^ u in X;

(c) if Un G /C n P is such that u^ ^ u in E, then u^ ^ u in X;

o

(d) for any u G dV \ KL, we have a{t, u) GP for t > 0.

Let A C X he di compact set in the X-topology such that A H 5 ^ 0, where

S = X\V, V = PU{-P).

Let B C E\D be closed and T be deflned as in Chapter 5. Deflne

X^ _ jr T r ( t , x) : [0,1] X X ^ X is continuous in the 1 •~ \ X-topology and T{t,V) cV j '

Thenr ( t , ^ ) = {1 - t)u e T*.

Theorem 9.2. Assume (Ai). Let A link B in the sense of Definition 5.1 and ^ satisfy the (PS) condition. Suppose that V is an admissible invariant set of ^ and that

(9.2) ao := sup^ < b^ := inf ^ . A B

Then ^ has a critical value defined by

(9.3) a* := inf sup ^{u). r^T^*r([o,i],A)n<s

Moreover, when 0 ^ KLa*, then JCa* nS^9ifa*> bo; JCa* nB^9ifa* = bo.

9.1. LINKING AND SIGN-CHANGING SOLUTIONS 197

Proof. For any F G T* we have that

F ( [ O , l ] , A ) n 5 ^ 0 , F ( [ O , l ] , A ) n 5 ^ 0 , F([0 ,1] ,A)CX

smd BnX CS. It fohows that F([0,1], A) n 5 n 5 ^ 0. Thus,

sup ^ r([o,i],A)n<s > sup ^

r([o,i],A)nSnB > inf ^

r([o,i],A)n<sn5 > inf ^

r([o,i],A)n5 > inf ^

which implies that a* > b^.

Case 1: In this case we assume that a* > h^. We suppose that /C^* n 5 = 0 and derive a contradiction. Note that for any u G P\{0}, the vector —^ (i )

o

points toward the interior of V- If ^ has no critical point on the boundary of o

P\{0}, then KLa* CV - By the (PS) condition, there are SQ > 0? o > 0 such that

l + | |$ ' (u) | | -<5o (9-4) ' II J " „ >^ for ue^-'[a*-eo,a*+eo]\{tCa^)so,

where {ICa*)5o := {u ^ E : dist£;(i^,/Ca*) < So}. By decreasing SQ, if necessary, we may assume that SQ < a^ —bo, and )C[a^ — £o,a^ -\- £o]nS = 9 (otherwise, it is trivial to get a contradiction via the (PS) condition). Let

Then {$'{u),V{u)) > 0 for all u and

(9.5) ($'(u),X^(u)) > ^ for any u G -^[a* - £o,a* + £o]\(/Ca05o-

Let

e o : = { u e £ ; : | $ ( u ) - a * | < 3£o},

ei:={ueE: | $ ( u ) - a * | < 2£o},

^, , ._ distg(M, 62) ^^^' '~ disti<;(u, Oi) + distij(u, 62) '


where B2 = E\So. Then ^{u)V{u) is a locahy Lipschitz vector field on both E and X since X is embedded continuously in E. We consider the following Cauchy initial value problem:

^ ^ ^ = -a<T{t, u))V{a{t, u)), a{0, u) = ueX,

which has a unique continuous solution cF{t^ u) in both X and E. Obviously,

(9.6) —^ y '' < 0.

By the definition of a* in (9.3), there exists a F G * such that

r([0,1], A) n 5 C ^«*+ô :={ueE: ^{u) < a* + So}.

Therefore, r([0,1], A) is a subset of ^«*+ô U V. Denote

A*:=r ( [0 , l ] ,A) .

We claim that there exists a TQ > 0 such that a{To,A*) C ^«*-ô/4 y p We consider three cases.

(i) If 1 G A* n P , then a{t, u) e V for all t >0. This is due to the assumption that V is an admissible invariant set.

(ii) If u e A'^û ^ (X:a*)35o,^ ^ ^- Since A* C ^«*+ô U P , we see that ^(i^) < a* +£o-

If ^(i^) < a* - £0, then ^(cr(t,i^)) < ^{u) < a* - e^ for ah t by (9.6).

If ^{u) > a* - £0, then u G ~^[a* - £o,«* +ô]- Note that

\\(j{t,u) -U\\< / ||(7(t,l^)||(it < t. Jo

o r

This implies that (j{t,u) ^ {â*)35o/2 for ^ ^ [0? ]• If there exists a o r o r

ti e [0, —^] such that ^{a{ti,u)) < a* - £0, then ^(cr(—^,i^)) < a* - SQ.

Otherwise, a* - £0 < ^{cr{t, u)) < a* + £0

or for SiWte [0,—^]. Then

Cr(t ,^) G ^ ^[a* - £ o , « * + ^ o ] \ ( ^ a * ) 3 5 o / 2


o r

for all t G [0, ]. It follows that ^{(j{t,u)) = 1 and

{$'{a{t,u)),V{a{t,u))}>eo/5o o r

for a lHG [ 0 , ^ ] . Therefore,

/ • ^ < a*+60- {^'ia{s,u),Via{s,u)))ds

Jo

< a - - .

Hence,

(9.7) ^(^(To,^)) < a* - y for any To > ^ .

(hi) If 1 G A*,!^ G {JCa*)35o,u ^ V, then ^{u) < a* + SQ. If moreover, ^(i^) < a* - So, then by (9.6), ^(cr(t, u)) < a* - So for ah t > 0. Assume that ^{u) > a* — £o- Then u G ^~^[a* — Soâ* -\- SQ]. If there exists a sequence {t^} and 0 ^ ^ such that a{tn,u) -^ ZQ in E, then cr has to travel at least (ô-units of time, and an argument similar to that of (ii) provides the proof.

o If there exist a sequence {tn} and ZQ G P such that a{tn,u) ^ ZQ in E and

o

therefore in X, then there exists a IN such that a{tN,u) G P . The remaining situation is when

(9.8) dist^; f cr([0, oo), ) , X:[a* - SQ, «* + ô]) := î > 0.

By the (PS) condition, there exists an e* > 0 such that

for

u G ~^[a* -£o ,«* +£o]\(^[«* - ^ o , « * +ô])5i-

Similarly, we suppose that ^{a{t,u)) > a* — SQ for all t. Then by (9.^

(9.10) (7( t ,^)G^-^[a*-£o,a*+ô]\(X:[a*-£o,a*+ô])5i .

Therefore,

or /^f^ (9.11) ^{a{^,u)) = ^{u)^ I (i^(cr(5,^)) < a * - 2 £ o .


By combining (9.7) and (9.11) for cases (ii)-(iii), we see that for any u G

A*,i^ ^ P , there exists a T^ > 0 such that (J(TU,U) G ^«*-ô/2y p gy

continuity, there exists an X-neighborhood Uu such that

^ ( T , , ^ n ) c ^ " * - ^ ° / ^ U P .

Since A* is compact in X, we get a TQ > 0 such that cr(ro, A*) C ^«*-ô/4y p We define

r a ( 2 r o 5 , ^ ) , 5 G [ 0 , ^ ] ,

r*(5,^) = < [ (7(ro,r(25-i,^)), 5G[^,i].

Then, T* G T*. If 5 G [0, ^] , we have that

r*(5, A) n 5 c cr(2ro5, A) n 5 c ^"° n 5 c ^«*-ô/4.

If 5 G [ ,1], then

r*(5,A)n5 c cr(ro,r(25-i,A))n5 c (7(ro,A*)n5 c (^"*-^°/ûP)n5

It follows that G(r*([0,1], A) n 5) < a* - £o/4, a contradiction.

Case 2: Assume a* = 6o- If â* H 5 = 0, then there are positive numbers ^15^2,^3 such that

(9.12) | | ^ ' ( ^ ) | |>£ i for | ^ ( ^ ) - a * | <£2 and dist£;(^, 5 ) < £3.

It holds true if we decrease £2 and £3. Thus, we may assume that £2 < £?£3/2(l+£i) . Let

(9.13) Mi:={ueE: dist^(^, B) < ^ , |^(^) - a*| < ^ }

and

(9.14) M2 := {^ G : dist^(^, B) < ^ , |^(^) - a*| < y } .

Then M2 ^ 0. In fact, if we choose F G * such that

sup ^(u) < a* +£2/3 , r([o,i],A)n<s


then we can find a i o G r ( [0 ,1] , A) n 5 n 5 ^ 0 since A finks B. Note tfiat since a* = 6o, we fiave

bo < ^{uo) < sup ^{u) < a* + £2/3, r([o,i],A)n<s

tfiat is, uo e M2 C Ml. Let

d is t£ ; ( i^ ,^ \Mi) eiM dist£;(i^, E\Mi) + dist£;(i^, M 2 ) '

and consider tfie Caucfiy initial value problem

[ ai{0,u) = u e E,

wfiicfi fias a unique continuous solution (Ji{t,u) in E. By (9.12)-(9.13), it is easily seen tfiat

Let u e ^«*+^2/3^

If tfiere is a t i < — sucfi tfiat cri{ti,u) ^ M2, tfien eitfier ^{cri{ti,u)) <

a* or dist£;(cri(ti,i^), 5 ) > —. Note tfiat

| |<Tl(t ,^t)-(Jl(t^^t) | | < | t - t ' | ,

tfius we fiave tfiat dist£;(cri(t,i^),5) > — , and fience, (Ti{tû) ^ B for all

t e [ 0 , | ] .

If a i ( i , u) e M2 for all t G [0, £3/4], then

£3

$ ( ( 7 i ( ^ , u ) ) = ^{u) + j ' d^{a,{t,u))

< a* ''^ ^^^ ' 3 4 ( 1 + e i )

< a - - .

Tfiat is, eitfier ^ ( c r i ( ^ , i ^ ) ) < a* - — or (Ji{t,u) ^ B for afi t G [0, y ] and

eacfi u e A^ E""*^^.


Next we show tha t ai{t,u) ^ B for dl\ u ^ A and t G [ 0 , ^ ] . First, cTi(£3/4,1^) ^ B. Further, by (9.15),

^2 nt

-L + ^ 1 Jo ^2 rt

< a* - / / î (cri (5 ,^))( i5 . -L + ^ 1 Jo

If cri(t,i^) G 5 , then ^{ai{t,u)) >ho=a*^ and we must have £î{(Ji{sû)) = 0

for 5 G [0,t]. This implies tha t (Ti{tû) ^ M2 and either ^{(Ti{tû)) < a*

or dist£;(cri(t,i^),5) > —. Both cases imply cri{t,u) ^ B. By the definition

of a*, we may choose a F i G ^* such tha t

sup ^ < a * + ^ . ri([o,i],A)n<s 3

For any i G A, if Ti{t,u) ^ S for some t, then Fi(t , i^) G P , since Fi(t,74) C X . By an argument similar to tha t used in the first case, we see tha t V is invariant under the negative gradient fiow ai generated by the vector field

• 3 ' 3

—îV. Therefore, cri( —,Fi ( t , i^ ) ) G V and hence, cri( —,Fi ( t , i^ ) ) ^ B, since

5 n P = 0. Let

F * ( t , ^ ) = <

( 7 i ( ^ , F i ( 2 t - l , ^ ) ) , ^ < t < l .

Then it is easy to check tha t F* G T*. But the above arguments also imply tha t F2([0,1], A) n 5 = 0, which contradicts the fact tha t A links B. This completes the proof of the theorem. D

9.2 Free Jumping Nonlinearities

We consider the semilinear elliptic boundary value problem

-Ai^ = / (x , i^ ) , in O,

(9.16) ; 1 = 0, on dfl,

9.2. FREE JUMPING NONLINEARITIES 203

where O C R ^ is a bounded domain with smooth boundary dft; f{x,t) is a Caratheodory function on O x R such that

(9.17) ' > a a.e. x ^ U as t

> b a.e. X ^ U as t -t

Let E be the Fucik spectrum (see Section 5.4 of Chapter 5 for details). Note that we only know the existence of the curves Cn and C12 in the square (A/_i, A/+i)^. It is quite a challenging problem to determine the Fucik spectrum and the position of (a, b) with respect to this spectrum. On the other hand, since the eigenfunctions associated with A/ (/ > 2) of —A are sign-changing, a natural problem is whether the solutions of (9.16) are sign-changing if resonance occurs around Xi {I > 2). In this section, we shall show

• how to deal with the case when (a, b) is not restricted to the inside of the square (A/_i, A/+i)^ described above, no matter whether or not resonance occurs;

• how to get more information concerning the solutions.

These results allow (a, b) to be independent of the Fucik spectrum. With some additional assumptions there is no need to involve the Fucik spectrum in dealing with problems with jumping nonlinear it ies.

(Bi) / G C\n X R), | /^(x,^) | < c(l + 1^1 -2) for a.e. x G O and ah u e K, where 2 < p < 2* and there exists a constant Co > 0 such that

f(x,t) — f(x, s) ^ w ^ t — s

(B2) a,b > Xk for some integer k > 2.

(B3) 2F{x,t) > Xk-it^ for ah (x,t) G O x R, where F{x,t) = J^ f{x,s)ds.

(B4) / (x,0) = 0 and 2F{x,t) < Kot'^ for all x e Q and \t\ < 5o, where 5o > 0, hio ^ (A/c-i,A/c) are constants.

f(x t)t — 2F(x t) (B5) liminf'^^ ' \ ^—^^ > c> 0 uniformly for x e Q; here a G (1,2)

is a constant.

By assumption (^2), the point (a, 6) may or may not lie on any curves Cn or C12 and may even lie outside the square (A/_i,A/+i)^ for all / > k. The


points a or 6 may be situated across multiple eigenvalues Xi {I > k). In particular, we permit a = b = Xi {Wl > k -\- 1). This means that resonance at infinity can occur at any Xi {I > k -\-l). Assumptions (^3) and (^4) contain

the case when lim — = Xh-i, a resonant case at the origin. Let Ei denote

the eigenspace corresponding to A/(/ > 1) and N^ = EiU - - - U E^. Define

(9.18) ^{u) := - / \Vu\'^dx - / F{x,u)dx, u G HI{Q).

We have

Theorem 9.3. Assume that f{x,t) satisfies (9.17) and that {Bi)-{B^) hold. Then equation (9.16) has a sign-changing solution u^ with ^{u^) > 0.

The next case includes double resonant, oscillating and jumping nonlin-earities.

(9.19) n ^ _^ ^^^^ ) ^^ ^ ^Q as t ^ ±00,

where A < b±{x) < A^+i (A: > 2).

Theorem 9.4. Suppose that (5i ) , (^3), (^4) and (9.19) hold. Assume that

(Be) min{6+(x),6_(x)} ^ Afc;

(B7) no eigenfunction of —A corresponding to X^ or A^+i is a solution of —Au{x) = b-^{x)u'^{x) — b-{x)u~{x).

Then equation (9.16) has a sign-changing solution u* with ^{u*) > 0.

Let E := HQ{Q) be the usual Sobolev space endowed with the inner product and norm

{u,v)= {\/U'\/v)dx, \\u\\ = ( \\/u\'^dx] , u^veE.

Let X := CQ{^) be the usual Banach space which is densely embedded in E. The solutions of (9.16) are associated with the critical points of the C^-functional

^{u) = -\\uf - f F{x,u)dx, u G Hl{Q). 2 Jn


By the theory of ehiptic equations, )C = {u e E : ^'{u) = 0} C X. The positive cones in E and X are given respectively by

PE := {u e E : u{x) > 0 for a.e. x G 0 }

and P := {u e X : u{x) > 0 for every x G O}.

It is well known that PE has an empty interior in E and P has a nonempty o

interior P= {u e X : u{x) > 0 for all x G ^^dyu{x) < 0 for all x G Ô}, o

where z/ denotes the outer normal. Therefore, P =P UdP. We rewrite the functional ^ as

Hu) = ^WuWl - ^ ( i (Co + l y + F(x,^))(ix,

/ \ 1/2

where ||i^||£; := ( /^(|Vi^p + (Co + l)\u\'^)dx] , which is equivalent to ||i^||.

Then the gradient of ^ at i is given by

^\u) = ^ - ( - A + (Co + l ) ) " ' ( / (x ,u) + (Co + 1)^) :=u- Ju,

where the operator J : E ^ E is compact and J{X) C X. In particular, by the strong maximum principle, J\x, the restriction of J to X, is strongly

o

order preserving; that is, for any u — v G P\{0}, we have Ju — Jv GP . Since / (x , 0) = 0, the ±P are invariant sets of the negative flow of the vector — ^ It is easy to check that V is an admissible invariant set.

Lemma 9.5. Assume that {Bi)-{B^) hold. Then ^ satisfies the (PS) condition.

Proof. Let {un} be a (PS) sequence, that is, ^\un) -^ 0,^(i^^) -^ c. By Theorem L41, it suflices to prove that {un} is bounded in E. In fact, by (^5),

there exists an i?o > 0 such that -f{x,t)t — F{x,t) > c\t\^ for all x G O and

\t\ > RQ. Because of (9.17), we may assume that \f{x,t)t\ < d? for x G O and \t\ > RQ. Then, for n sufliciently large, we have the following estimates:

c+ ll nll

= ( / + / )(-f{x,Un)Un- F{x,Un))dx

> —C-\- C / \Un\^dx.


Choose Co = (2 - a){N + 2)/(2AT + 4 - Na). Then CQ G (0,1) and

/

<-{

\Un\'^dx ,\>Ro

J\u^\>Ro ^ ^

„ 2CQN • / (2Ar + 4) \ 2JV + 4

< ( c + c||w„||)^^^^||w„f^».

Consequently,

\\Un\\ = {^'{Un),Un) ^{j + / )f{x,Un)Undx

< \\Un\\ -^C^C \Un\ dx '\u^\>Ro

< \\UJ^C^{C^C\\UJ)'^' '°^/"| |^nf^°.

Since a G (1, 2) and CQ G (0,1), we have that 2(1 — co)/cr + 2co < 2. Thus, we see that {||i^n||} is bounded. D

Rewrite ^ as

Hu) = huf-U\u-\\l-h\\u+g-fp{x,u)dx, ueH'om, J S2

where

H{x,u) := / h{x, t)dt; h{x, t) = / (x , t) - {bt^ -at~). Jo

Let El denote the eigenspace of A/(/ > 1) and Nk = EiU - - - U Ek- Then

Lemma 9.6. Assume {B2). Then ^{u) -^ —00 as \\u\\ -^ oo,u ^ N^.

Proof. Without loss of generality, we assume that a > b. For u = U- -\-uo G Nk with U- G Nk-i,uo G Ek, then

* N = hwf - U\u-\\l - h\\u+\\l - f H{x,u)dx. J S2


We have that

* W <l\\uf-l{a-b)\\u-\\l-lb\\u\\l- [ H{x,u)da Jn 2" " 2 ' '" " 2

< i ( l - - ^ ) | | ^ _ f + i(l-A)||^,||2

{a — b)\\u II2 — / H{x,u)dx Q

1 ~2

^ ^k-l ^ ^k-1 ^k

-ha-b)\\u-g- f Hix,u)dx

Therefore, there exists an e > 0 such that

^{u) < -s\\u\\^ - I H{x,u)di

h(x,t) , , , ^(u) = 0, thus we have hm sup -—-

\t\ôo t IklHoo ll'^ll which implies the conclusion of the lemma. D

for all u G Nk' Recall that lim — = 0, thus we have limsup -—— < —e, 1.1 J- M ^ . M ^

Lemma 9.7. Suppose that (9.17) (or (9.19) ) and {B4) hold. Then there exist po > O5 Co > 0 such that ^{u) > CQ for u G Nj^_-^ with \\u\\ = po-

Proof. By (9.17) (or (9.19)) and (^4), we may choose a 5i > A so large that 2F(x,t) < sit'^ for \t\ > 5o, x G O, where 5o comes from (^4). Choose 52 : = 2 ^ 1 . T h e n ,

(9.20) 2F{x,t) < S2t^ - sisl for \t\ >so,xe n.

For any u G Nj^_-^, we write u = v -\-w with v e E^ ® Ek+i © • • • © Ei-i and

^— + 1051 + ^ Xk — 1^0 ^0

w G iV/"ii, where / is large enough so that Xi > h IO51 H . Let

/ o o i A (52 + A / ) 2 , (Afe + ^ o ) 2 77^ , N

(9.21) /ii := ;j w -\ V — F{x,v-\-w).


If I'U +1(;| < 5o, then by condition (^4) and the choice of A/, we have that

/il > W + V --K.o{v^w)

. (52 + A/) - 2/^0 2 , (Afe + ^0) - 2/^0 2 I I (9.22) ^ i ^ + 4 ^ -ô l^^ l

^ ^((g2 + Az-2/ô)(Afe-/ô))V^ V

> 0.

If I'U +1(;| > 5o, then by (9.20), we get that

(9.23) /il > /i2 + /i3,

where

/n o/i\ A/ - 52 2 , (Afe - ^0) 2 (9.24) /i2 := — z — ^ + ^ ~A ^ ~ n^vw,

o 4

(9.25) /is := — ^ — ^ ^ V - (52 - tô)vw + ^ - ^ .

We claim that /i2 is always greater than or equal to zero. In fact, if 1 ;!

K.o\w\ > 0, then

(9.26) M2 > ^^^w^ + ( ^ ^ | « | - ôH)\v\ > 0.

Otherwise, by the choice of A/, we have that

/A/ —52 4:Hin , 9 Xk —1^0 9 (9.27) ^2 > ( ^ ^ - l - ^ ) « ^ ' + ^ ^ « ' > 0.

Also, we have that

. (Az + 5 2 ) - 2 5 2 2 I 1/ M I , ' I' O /is > ^ ^ - \S2 - f^0\{\v\ ^ \w\)\v\ ^ ^—

( A z + 5 2 ) - 2 5 2 2 / M l

-(^2-^0)H^ + ^

^ (A/ - 552 + 4/ô) 2 ^("^2 - ^0) 2 I ' I' O

^ 3(52 - no) 2 I ^ 2

Set

(9.28) Qi := {x en:\v^w\< 5o}, O2 := {x G O : 1 ; +1(;| > 5o}.


Since dim.Ni_i < oo, we may find a constant C/_i such that

(9.29) max|i;| < Ci_i\\v\\ foidllv e Ni_i.

Let

(9-30) So := —^ r T 7 ^ ( l - T^)-8(52 - /^0)Cf_/ Xk^

Then (o > 0. By (9.22)-(9.28), we have

/ /jîdx = / /jîdx -\- / /jîdx

If meas02 > SQ, then

> / fiidx JQ2

2 JQ2 2

3(52 - ô)|| ||2 , Sl4 (9.31) ^ M x > _ - v - ^ ^ ' - ^ | | ^ f + r ^ ^ , .

If meas02 < ( o, then by (9.29)-(9.30),

(9.32) f fiidx > - ^ ^ ^ ^ ^ ^ C f _ i | | i ; f m e a s 0 2

Combining (B4) and (9.20)-(9.27), we have

^(u) = l{\\vf+\\wf)-1 F{x,v + w)dx

> \\\vf + \\\wf + \Xk\\v\\l + \xi\\w\\l - jj{x,u)da

> \{l-^J\\vf + \{l-'p\wf + J^f^,dx

> l n i i n { ( l - g ) , ( l - g ) } | H | 2 + ^^;.idx

> \{l-j^)\\uf+ f ^^ldx.


By (9.31), if meas O2 > ( o, then

(9.34) û) > i ( l - g ) | H | 2 - f c - ^ | H | 2 + f ^ 5 o

(9.35) > i ( l - g ) | H | 2 - f c - ^ | H | 2 + f^<5o.

By (9.32), if meas O2 < ( o? then

(9.36) û) > \{1 - g ) | | ^ . f - A ( i _ g ) | | , | | 2 > _L(i _ g ) | | „ | | 2 .

By (9.34)-(9.36), we may find po > 0 and CQ > 0 such that ^{u) > CQ for u G Ni^_^ with ll ll = po. •

Proof of Theorem 9.3. Invoking condition (^3), we readily have ^(u) < 0 for all u G Nk-i. By Lemmas 9.5-9.7, there exist RQ > po > ^ such that

ao := sup^(i^) < 0 < Co < 60 •= inf ^(i^), A B

where

A:={u = v^syQ:ve Nk-i.s > 0, ||^|| = Ro} U [A^ -i n BR,],

B:={ue N^_, : ||^|| = po}

and yo G Ek satisfying \\yo\\ = 1- Theorem 9.2 implies that there is a critical point i * satisfying ^^(i^*) = 0, ^(i^*) = a* > 60 > 0- Obviously, 1 * ^ 0 and either i * G 5 or i * G 5 . The second alternative occurs when ^{u*) = bo := inf^ ^{u). Both cases imply that u* is sign-changing. D

Lemma 9.8. Under the hypotheses of Theorem 9.4, ^{u) -^ —00 for u e N^ as \\u\\ -^ 00.

Proof. Note that

Hu) = ^\\uf- f {^h^{x){u^f ^h_{x){u-f ^ H{x,u))dx, ueE,

where

H (xû) := / h{x,t)dt; h{x,t) = f{x,t) — (b-^{x)t~^ — b-{x)t j .

Note that min{6+(x), 6_(x)} > and ^ A , and recall the variational characterization of eigenvalues {A^}. We then have the following estimates for any


ueNk.

= lhf- [ H{x,u)dx

- \ { \ +[ )(b+{x){u+f + b.{x){u-f)dx '^^Jb-{x)>b+(x) Jb-(x)<b+(x)'^ ^ '

-\W--^\ b+ix)u'dx

(b-{x) — b-^{x)]{u )'^dx (x) ^ ^

1

2 Jb_(x)>b+

- / h-{x)v?dx 2 Jb_ix)<b+ix) 1

lb-^{x) — b-{x)]{u'^)'^dx — / H{x,u)dx (x) ^ ^ Jn 2 Jb_(x)<b+

<h\uf- ^ I b^{x)u^dx -I 2 Jb-{x)>b+{x)

2 ^ Jb-{x)>b+{x)

-- / b-{x)v?dx — I H{x,u)dx 2 Jb-{x)<b+{x)

<i|HI^ , - / inm{b-^{x),b-{x)}u^dx — / H{xû)di 2 J Q J Q

< - ( 5 | | ^ f - / H{x,u)dx;

the lemma follows immediately.

P r o o f of T h e o r e m 9.4. By Lemma 9.8 and (^s ) , there exist RQ > po > 0 such tha t

ao := sup^(i^) < 0 < Co < 6o •= inf ^(i^), A B

where A and B are defined as in the proof of Theorem 9.3. To apply Theorem 9.2, we just have to check the (PS) condition. Assume tha t {un] is a (PS) sequence. Otherwise, we may assume tha t ||i^^|| -^ oo as n -^ oo. Let w^ = '^n/ll'^nll- Then ||t^n|| = 1 and, for a renamed subsequence, Wn ^ w weakly in E, strongly in 1/^(0) and a.e. in O. Moreover,

{^\Un),v) = {Un,v) - / f{x,Un)vdx ^ 0

and f f{x,Un)v

[Wn^v) — / ax ^ 0.


By (9.19), we see tha t —Aw = b-^w~^ — b-W~. Since

we see tha t

/ (b^{w^f ^b_{w-fyx=1.

This implies tha t w ^ 0. Let w = W- -\- W-^ with W- G Nk^W-^ G A^^ and w = !(;+ — W-. Take g'(x) = &+(x) when w{x) > 0 and q{x) = b-{x) when i(;(x) < 0. Then we have tha t —Aw = q{x)w. Hence

Ih+f-

It follows that

- I k - f

0 <

<

=

<

= / q{x){w^f - / q{x){w_) Jn Jn

\\w+f - Xk+^\\w+\\l

Jn

\\w.f- 1 q{x){w.f Jn

Ww.f-Xk f{w-f

< 0.

Tha t is, | | i ^±P = J^q{x){w±)'^. The only way this can happen is q{x) = A^ when W-{x) ^ 0 and q{x) = A^+i when W-^{x) ^ 0, and therefore, either W-is an eigenfunction of A^ or W-^ is an eigenfunction of Afc+i. This implies tha t W-{x)w-^{x) = 0. Thus q{x) = 6+ = A^ when W-{x) > 0; q{x) = b- = Xk when W-{x) < 0; q{x) = 6+ = A^+i when W-^{x) > 0; q{x) = 6_ = A^+i when w^{x) < 0. Then

—Aw- = XkW- = 6+i(;l — b-wZ,

—Aw-^ = A/c+ii(;+ = b-^w^ — b-W^.

Hence, w± = 0. This is a contradiction and the (PS) condition follows. We obtain the conclusions of Theorem 9.4 as in the proof of Theorem 9.3. D

N o t e s and C o m m e n t s . The assumptions in (9.19) mean tha t the problem is double resonant. An earlier paper on this line is H. Berestycki-D. G. deFigueiredo [57] (see also M. F . Furtado-L. A. Maia-E. A. B. Silva [166]). At the end of the Chapter 5, we gave historical notes and our comments.


Their results are closely related to the Fucik spectrum. They did not make the stronger assumptions on ^{u) that are made here. Sign-changing solutions have attracted much attention in recent years (see T. Bartsch [29], T. Bartsch-K. C. Chang-Z. Q. Wang [31], T. Bartsch-Z. Q. Wang [42], A. Castro-J. Cossio-J. M. Neuberger [83, 84], A. Castro-M. Finan [85], E. N. Dancer-Y. Du [126], S. Li-Z. Q. Wang [218, 217], Z. Q. Wang [374]). Also there are some interesting results in G. Chen-W. Ni-J. Zhou [100], Z. Ding-D. G. Costa-G. Chen [137], D. G. Costa-Z. Ding-J. Neuberger [108], J. Neuberger [266] and J. Neuberger-J. W. Swift [267] for numerical methods for sign-changing solutions. These have suggested various types of sign-changing solutions. In T. Bartsch [29] (see also T. Bartsch-Z. Q. Wang [42]), the author established an abstract critical theory in partially ordered Hilbert spaces by virtue of critical groups and studied superlinear problems. In S. Li- Z. Q. Wang [217], a Ljusternik-Schnirelman theory was established for studying the sign-changing solutions of an even functional. Some linking type theorems were also obtained in partially ordered Hilbert spaces. We also refer readers to T. Bartsch-Z. Liu-T.Weth [38] for the existence of sign-changing solutions. Concerning the theory of ordered Banach spaces, we refer to the paper of H. Amann [8]. As for the properties of flow invariant sets, see H. Brezis [65] and K. Deimling [130]. Theorems of this section were given in M. Schechter-Z. Q. Wang-W. Zou [331]. The readers may also consult T. Bartsch-Z. Q. Wang [44] (on superlinear Schrodinger equations), T. Bartsch-T. Weth [46, 47] (on superlinear elliptic equations and singularly perturbed elliptic equations), E. N. Dancer-S. Yan [127] (on sign-changing mountain pass solutions), M. Schechter-W. Zou [333] (on asymptotically linear Schrodinger equations), Z. Zhang-S. Li [382], W. Zou [390](on sign-changing saddle point) and W. Zou [388] for more results.

Documents

Sign-Changing Solutions 9 Sign-Changing Solutions In this chapter, we study the existence of sign-changing solutions for nonlinear elliptic equations via linking methods. A linking