JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS ARTICLE NO.
223, 397]417 Ž1998.
AY985938
Solutions for Resonant Elliptic Systems with Nonodd or Odd Nonlinearities* Wenming Zou† Department of Applied Mathematics, Tsing Hua Uni¨ ersity, Beijing 100084 and Institute of Mathematics, Academia Sinica, Beijing 100080, People’s Republic of China Submitted by John La¨ ery Received March 18, 1997
We consider by minimax techniques a class of cooperative elliptic systems at resonance. If the nonlinear term is nonodd, we establish some new existence theorems about Žnontrivial. solutions. If the nonlinearity is odd, we obtain existence of infinitely many solutions, a topic about which very little is known. Q 1998 Academic Press
Key Words: Resonance; cooperative elliptic systems; nonodd nonlinearity; odd nonlinearity
1. INTRODUCTION In this paper we study nonlinear elliptic systems of the form
¡yDu s lu q d ¨ q f Ž X , u, ¨ . ¢u s ¨ s 0
~yD¨ s d u q g ¨ q g Ž X , u, ¨ .
in V in V on V ,
Ž P.
where V is a bounded smooth domain in R N and l, g , d g R. The nonlinearities Ž f, g . are the gradient of some function, that is, there exists a function F g C 1 Ž V = R 2 , R. such that =F s Ž f, g ., the so-called cooperative case. System ŽP. is called resonant if s Ž A*. l s ŽyD . / B, where A* s
ž
l d
d ; g
/
s Ž A* . s j , z 4
* Supported by the Chinese National Science Foundation. † E-mail:
[email protected]. 397 0022-247Xr98 $25.00 Copyright Q 1998 by Academic Press All rights of reproduction in any form reserved.
398
WENMING ZOU
denotes the spectrum of the matrix A* and s ŽyD . s l k : k s 1, 2, . . . , and 0 - l1 - l 2 - ??? 4 denotes the eigenvalues of the Laplacian on V with zero boundary condition. First we point out that problem ŽP. has been considered by Costa and Magalhaes ˜ w1x. In w1x the variational structure was established, and some results on the existence of solutions were obtained by minimax techniques under a so-called condition of nonquadraticity at infinity, but the solutions were not necessarily nontrivial, and the authors did not consider the case of odd nonlinearities. For related problems, including noncooperative elliptic systems, the reader can consult the references in w1, 2x. In this paper we will consider systems ŽP. that are resonant. Our goal is twofold: if the nonlinearity =F is nonodd, we will establish some new theorems on the existence of Žnontrivial. solutions; if the nonlinearity =F is odd, we will introduce some new conditions that allow us to prove the existence of infinitely many solutions. To the best of my knowledge, very little is known about odd nonlinearity. In Section 2, we deal with the nonodd case. For this purpose, we make the following assumptions: ŽZ 1 . There exist constants C0 ) 0, 0 - a - 1 such that =F Ž X , U . F C0 Ž < U < a q 1 .
for U g R 2
and for a.e. X g V .
ŽZ 2 . There exist functions u 1 , u 2 g CŽ V, R. such that
u 1 Ž X . F lim inf
=F Ž X , U . ? U
< U <ª`
1q a
F lim sup < U <ª`
=F Ž X , U . ? U < U < 1q a
F u2Ž X .
uniformly for a.e. X g V, where HV u 1Ž X . / 0, HV u 2 Ž X . / 0, the dot denotes the inner product in R 2 and < ? < denotes the norm in R 2 . We first prove the following result. THEOREM 1.1. Assume ŽZ 1 . and ŽZ 2 . with u 1Ž X . G 0 for a.e. X g V. Then system ŽP. has a solution. If we make additional assumptions on F near the origin, we can obtain the existence of a nontrivial solution. More precisely, if we assume that s Ž A*. s j , z 4 with j s l k Ž k G 2., l j - z F l jq1 Ž l0 s y`. and ŽZ 3 . There exist constants d 1 and d 2 with max l j y z , l ky1 y l k 4 F d 1 F d 2 - 0 such that
d 1 F inf
U/0
2FŽ X, U .
2
,
lim sup Uª0
2FŽ X, U .
F d2
SOLUTIONS FOR RESONANT ELLIPTIC SYSTEMS
399
uniformly for a.e. X g V, then we have THEOREM 1.2. Suppose that ŽZ 1 ., ŽZ 2 ., and ŽZ 3 . hold, where F Ž X, 0. s 0 and u 1Ž X . G 0 for a.e. X g V. Then system ŽP. has at least one nontri¨ ial solution. In Section 3, we deal with system ŽP. with an odd nonlinear term. We assume ŽO 1 . =F Ž X, yU . s y=F Ž X, U ., F Ž X, 0. s 0 for a.e. X g V and for U g R 2 . ŽO 2 . lim inf U ª 0 2 F Ž X, U .r< U < 2 s q` uniformly for a.e. X g V. ŽO 3 . lim
5 U 5 2E s 5 u 5 2 q 5 ¨ 5 2
for U s Ž u, ¨ . and F s Ž w , c . in E. Let © e1 s Ž x 1 , x 2 . and © e2 s Ž y 1 , y 2 . g © © 2 R be the normalized eigenvectors of A* such that A*e s j© e1 , A*e 1 2s © © © © © q y z e2 , and e1 ? e2 s 0, < e1 < s < e2 < s 1. For any a g R, let Ha , Ha , and Ha0 be the subspaces of H01 Ž V ., where the quadratic form u ª 5 u 5 2 y a 5 u 5 22 is positive definite, negative definite, and zero, respectively. Here and throughout this paper, we use 5 ? 5 p to denote the norm of L p Ž V .. Let N s Hj0 = Hz0 ,
V s Hjy = Hzy,
W s Hjq = Hzq.
400
WENMING ZOU
Set A1 s id y j ŽyD .y1 , A 2 s id y z ŽyD .y1 , where id denotes the identity from H01 Ž V . to H01 Ž V .. We introduce the operator A: E ª E, A s Ž A1 , A 2 ., defined by A Ž U . s Ž A1 u, A 2¨ .
for any U s Ž u, ¨ . g E.
A is a bounded self-adjoint linear operator from E to E and Ker A s N with dim N - `. The space E splits as E s V [ N [ W, where V, W are invariant under A, A < V is negative, and A < W is positive definite. Consider the following functional: I Ž U . [ 12 ² AU, U :E y J Ž U . ,
J ŽU . s
HVF˜Ž X , U . ,
U s Ž u, ¨ . g E,
© © . Ž . where F˜Ž X, s, t . s F Ž X, se 1 q te 2 . Then the solutions of system P are 1 the critical points of the C functional I: E ª R. Now we are ready to state the main results about the existence of infinitely many solutions of system ŽP. when there is symmetry.
THEOREM 1.3. Suppose that F satisfies ŽO 1 ., ŽO 2 ., ŽO 3 ., and ŽO4 .. Then system ŽP. has infinitely many solutions Un4`ns1 with I ŽUn . ª 0y as n ª `. THEOREM 1.4. Suppose that F satisfies ŽO 1 ., ŽO 2 ., ŽOX3 ., and ŽOX4 .. Then system ŽP. has infinitely many solutions Un4`ns1 with I ŽUn . ª 0y as n ª `. As an immediate consequence, we have the following result for a single elliptic equation at resonance. COROLLARY 1.1. Consider the equation
½
yDu s l k u q f Ž X , u . us0
in V on V .
Ž P9 .
Assume that f Ž X, t . is odd in t and lim inf t ª 0 2 F Ž X, t .rt 2 s q`, uniformly for a.e. X g V, where F Ž X, t . s H0t f Ž X, s . ds. In addition, assume that there exist R 0 ) 0 and M0 such that one of the following conditions holds: Ži. lim < t < ª` f Ž X, t . s 0 uniformly for a.e. X g V and F Ž X, t . F 0 for a.e. X g V and for < t < G R 0 . Ži.9 < f Ž X, t .< F M0 for a.e. X g V and for t g R and F Ž X, t . y 1 Ž f X, t . t G hŽ X . for a.e. X g V and for < t < G R 0 , where h g CŽ V, R. with 2 HV hŽ X . ) 0. Then the resonant problem ŽP9. has infinitely many solutions u n4`ns1 such that I Ž u n . ª 0y as n ª `, where I Ž u. s 12 HV < =u < 2 y 1 2 Ž . 2 l k HV u y HV F X, u .
401
SOLUTIONS FOR RESONANT ELLIPTIC SYSTEMS
Remark 1.2. As we have pointed out, very little is known about the existence of infinitely many solutions for resonant single elliptic equation and elliptic systems Žboth cooperative and noncooperative.. The recent paper w3x considered a strongly resonant single elliptic equation with odd nonlinearities, but only obtained a finite number of solutions. For nonresonant noncooperative elliptic systems, one can consult Bartsch and Clapp w11x for a result on infinitely many solutions in the superquadratic case. In our present paper, we shall expand the methods used in w3x. The organization of this paper is as follows. In Section 2, we consider the nonodd case; it contains some lemmas and the proofs for Theorems 1.1 and 1.2. Section 3 also contains some lemmas, and Theorem 1.3, Theorem 1.4, and Corollary 1.1 will be proved. 2. =F IS NONODD We begin by establishing the following auxiliary result. LEMMA 2.1. Assume ŽZ 1 . and ŽZ 2 .. If a sequence Un s Un0 q Unq q E s N [ W [ V with Un0 g N, Unq g W, Uny g V is such that 5 Un 5 E ª ` and ŽUnqq Uny .r5 Un 5 E ª 0 as n ª `, then
Unyg
u 1Ž X . G 0 for a.e. X g V implies that
Ži.
lim inf nª`
Žii.
HV
=F Ž X , Un . ? Un a 5 Un 5 1q E
) 0,
lim inf nª`
HV
F˜Ž X , Un . a 5 Un 5 1q E
) 0.
u 2 Ž X . F 0 for a.e. X g V implies that
lim sup nª`
HV
=F Ž X , Un . ? Un a 5 Un 5 1q E
- 0,
lim sup nª`
HV
F˜Ž X , Un . a 5 Un 5 1q E
- 0,
© © . where F˜Ž X, s, t . s F Ž X, se 1 q te 2 .
˜ and conditions Proof. Case Ži.. Some computation, the definition of F, ŽZ 1 . and ŽZ 2 . imply that < =F˜Ž X, U .< F C1Ž< U < a q 1. for a.e. X g V and for U g R 2 , where C1 ) 0, and u 1 Ž X . F lim inf < U <ª`
=F˜Ž X , U . ? U
1q a
F lim sup < U <ª`
=F˜Ž X , U . ? U < U < 1q a
F u2Ž X . .
Furthermore, N s Hj0 = Hj0 satisfies the unique continuation property, that is, if U g N and UŽ X . s 0 on a set of positive measure, then
402
WENMING ZOU
UŽ X . s 0 for a.e. X g V. Moreover, dim N - ` implies that there exists C2 ) 0 such that < U 0 < F sup U 0 Ž X . : X g V 4 F C2 5 U 0 5 E
for all U 0 g N.
Using the same techniques as in the proof of Lemma 3.2 in w5x, we can prove that, for any « 1 ) 0 and « 2 ) 0, there exist d Ž « 1 . ) 0 and d Ž « 2 . ) 0 such that mes X g V : U Ž X . - d Ž « 1 . 5 U 5 E 4 - « 1
for any U g N _ 0 4 ,
mes X g V : U Ž X . ) d Ž « 2 . 5 U 5 E 4 - « 2
for any U g V [ W.
If we set V 1 n [ X g V : Un0 Ž X . G d Ž « 1 . 5 Un0 5 E 4 , V 2 n [ X g V : Unq Ž X . q Uny Ž X . F d Ž « 2 . 5 Unqq Uny 5 E 4 , then mesŽ V _ V 1 n . - « 1 , mesŽ V _ V 2 n . - « 2 , and hence V 1 n l V 2 n / B if « 1 and « 2 are chosen small enough. Therefore, Un Ž X . 5 Un 5 E
G G
Un Ž X . 5 Un 5 E
F F
Un0 Ž X . y Unq Ž X . q Uny Ž X . 5 Un 5 E
d Ž « 1 . 5 Un0 5 E y d Ž « 2 . 5 Unqq Unq 5 E ª d Ž «1 . , 5 Un 5 E Un0 Ž X . q Unq Ž X . q Uny Ž X . 5 Un 5 E C2 5 Un0 5 E q d Ž « 2 . 5 Unqq Uny 5 E 5 Un 5 E
ª C2 ,
for X g V 2 n l V 1 n as n ª ` and Un Ž X . 5 Un 5 E
F
Un0 Ž X . q Unq Ž X . q Uny Ž X .
F d Ž «1 .
5 Un 5 E 5 Un0 5 E 5 Un 5 E
for X g V 2 n _ V 1 n as n ª `.
q d Ž «2 .
5 Unqq Uny 5 E 5 Un 5 E
ª d Ž «1 . ,
403
SOLUTIONS FOR RESONANT ELLIPTIC SYSTEMS
On the other hand, for any « 3 ) 0, there exists C3 ) 0 such that
Ž u 1Ž X . y « 3 . < U < 1q a y C3 F =F˜Ž X , U . ? U F Ž u 2 Ž X . q « 3 . < U < 1q a q C3 for any U g R 2 and for a.e. X g V. Now, for any « ) 0, combining the above arguments and letting n be large enough, we have the following estimates. We denote by c various positive constants.
HV
=F˜Ž X , Un . ? Un a 5 Un 5 1q E
1n lV 2 n
G
G
HV HV
1n lV 2 n
1n lV 2 n
Ž u 1Ž X . y « 3 . u 1Ž X .
a 5 Un 5 1q E
< Un < 1q a a 5 Un 5 1q E
G Ž d Ž «1 . y « .
1q a
G Ž d Ž «1 . y « .
1q a
y c« 3 y
< Un < 1q a
HV
y «3c y
1n lV 2 n
žH
C3 mes V
y
a 5 Un 5 1q E
c a 5 Un 5 1q E
u 1Ž X . y c« 3 y
u 1Ž X . y
V
HV_V
c a 5 Un 5 1q E
u 1Ž X . y 1n
HV_V
u2Ž X . 2n
c a 5 Un 5 1q E
G Ž d Ž «1 . y « .
1q a
ž
HVu Ž X . y c« 1
1
y c« 2 y c« 3 y
/
c a 5 Un 5 1q E
This implies the following lower limit: lim inf nª`
HV
=F˜Ž X , Un . ? Un 1n lV 2 n
a 5 Un 5 1q E
G Ž d Ž «1 . y « .
1q a
žH
u 1Ž X . y c« 1 y c« 2 y c« 3 .
V
/
Similarly, we have the following estimates about the upper limit: lim sup nª`
HV
=F˜Ž X , Un . ? Un 2 n _V 1 n
F lim sup nª`
HV
a 5 Un 5 1q E
Ž u 2 Ž X . q « 3 . < Un < 1q a q C3 2 n _V 1 n
a 5 Un 5 1q E
.
/
404
WENMING ZOU
F lim sup nª`
d Ž «1 . q « .
žŽ žŽ
1q a
F lim sup c d Ž « 1 . q « . nª`
F Ž d Ž «1 . q « .
1q a
HV
1q a
2 n _V 1 n
Žu2Ž X . q «3. q
mes Ž V _ V 1 n . q
c a 5 Un 5 1q E
c a 5 Un 5 1q E
/
/
«1c
and
lim sup nª`
HV_V
=F˜Ž X , Un . ? Un a 5 Un 5 1q E
2n
F lim sup nª`
HV_V
Ž u 2 Ž X . q « 3 . < U < 1q a q C3 2n
a 5 Un 5 1q E
F c« 2 .
Combining the above arguments, we have
lim inf nª`
HV
=F˜Ž X , Un . ? Un a 5 Un 5 1q E
G Ž d Ž «1 . y « .
1q a
žH
u 1Ž X . y c« 1 y c« 2
V
y c« 3 y Ž d Ž « 1 . q « .
1q a
/
« 1 c y c« 3 .
Noting that « , « 2 , and « 3 are chosen arbitrarily, we obtain lim inf nª`
HV
=F˜Ž X , Un . ? Un a 5 Un 5 1q E
G Ž d Ž «1 . .
1q a
žH
u 1 Ž X . y c « 1 ) 0,
V
/
since « 1 is also arbitrary. To prove the second conclusion of case Ži., we first note that condition ŽZ 1 . and the definition of F˜ imply the following growth conditions: F˜Ž X , U . F C4 Ž < U < 1q a q 1 .
for any U g R 2
and for a.e. X g V ,
where C4 ) 0 is a constant. On the other hand, by ŽZ 2 ., we have that, for any « ) 0, there exists C5 ) 0 such that
Ž u 1Ž X . y « . < U < 1q a y C5 < U < F =F Ž X , U . ? U F Ž u 2 Ž X . q « . < U < 1q a q C5 < U <
405
SOLUTIONS FOR RESONANT ELLIPTIC SYSTEMS
for all U g R 2 . It follows that
Ž u 1Ž X . y « . r a Ž s 2 q t 2 .
Ž1q a .r2
y C5 Ž s 2 q t 2 .
1r2
F sf Ž X , sr , tr . q tg Ž X , sr , tr . F Žu 2 Ž X . q « . r a Ž s2 q t 2 .
Ž1q a .r2
q C5 Ž s 2 q t 2 .
1r2
,
for Ž s, t . g R 2 and for r g Ž0, 1.. Noting that F Ž X , s, t . s s
1
H0
dF Ž X , sr , tr . dr
q F Ž X , 0, 0 .
1
H0 Ž sf Ž X , sr , tr . q tg Ž X , sr , tr . . q F Ž X , 0, 0. ,
we have
Ž u 1Ž X . y « . 1qa
< U < 1q a y C5 < U < y c
F FŽ X, U . F
Žu2 Ž X . q « . 1qa
< U < 1q a q C5 < U < q c,
for any U g R 2 and for a.e. X g V. Hence
u 1Ž X . 1qa
F lim inf < U <ª`
FŽ X, U .
1q a
F lim sup < U <ª`
FŽ X, U .
1q a
F
u2Ž X . 1qa
.
Furthermore, by simple computation, we have
u 1Ž X . 1qa
F lim inf < U <ª`
F˜Ž X , U .
1q a
F lim sup < U <ª`
F˜Ž X , U .
1q a
F
u2Ž X . 1qa
,
uniformly for a.e. X g V. Now, using the same arguments as in the proof of the first conclusion, it is easy to prove that lim inf nª`
HV
F˜Ž X , Un . a 5 Un 5 1q E
) 0.
Case Žii. The proof of case Žii. is similar to the proof of case Ži.; we omit the details. The proof of the lemma is now complete.
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WENMING ZOU
Based on the above lemma, we show how conditions ŽZ 1 . and ŽZ 2 . imply the compactness condition ŽC.c in the Cerami sense Žcf. w8x.: Any sequence Un4 ; E such that I ŽUn . ª c and Ž1 q 5 Un 5 E .5 I9ŽUn .5 E ª 0 possesses a convergent subsequence. It was shown in w5x that condition ŽC.c actually suffices to get a deformation theorem and then, by standard minimax arguments, it allows rather general minimax results Žcf. w1, 2, 8, 9x.. LEMMA 2.2. If F satisfies ŽZ 1 . and ŽZ 2 . with u 1Ž X . G 0 Ž or u 2 Ž X . F 0. for a.e. X g V, then I satisfies the compactness condition ŽC.c for e¨ ery c g R. Proof. We suppose that the first alternative Ži.e., u 1Ž X . G 0 for a.e. X g V . holds. The proof with the second alternative is similar. Assume Un4 is such that I ŽUn . ª c and Ž1 q 5 Un 5 E .5 I9ŽUn .5 E ª 0 as n ª `. Since =F and hence =F˜ have subcritical growth, it is enough to prove that Un4 is bounded in E. Assume that 5 Un 5 E ª `, and write Un s Un0 q Unq q Uny g N [ W [ V s E. Since A < V is negative definite and A < W is positive definite, by combining condition ŽZ 1 ., the Holder ¨ inequality, and the Young inequality, we have 5 Unqq Uny 5 2E F c 5 Unqq Uny 5 E q c 5 Un0 5 E q c. It follows that Unq 5 Un 5 E
ª 0,
Uny 5 Un 5 E
ª0
as n ª `.
Now we make the following claim: There exists d ) 0 such that ² I9ŽUn ., Un0 :E F yd for n sufficiently large. Using this claim, we get a contradiction: <² I9ŽUn ., Un0 :E < F Ž1 q 5 Un 5 E .5 I9ŽUn .5 E ª 0. Hence the compactness condition holds. Finally, to finish the proof of this lemma, it is sufficient to prove the claim above. In fact, if the claim is not true, then for dn s 1rn, there exists a subsequence, say Un4 , such that y
1 n
F ² I9 Ž Un . , Un0: E s ² AUn , Un0 :E y
HV=F˜Ž X , U . ? U n
hence, lim sup nª`
HV
=F˜Ž X , Un . ? Un0 a 5 Un 5 1q E
F 0.
0 n ,
407
SOLUTIONS FOR RESONANT ELLIPTIC SYSTEMS
On the other hand, we have
HV
=F˜Ž X , Un . ? Ž Unqq Uny . a 5 Un 5 1q E
F
HV
Fc
c Ž 1 q < Un < a . Ž < Unq< q < Uny< . a 5 Un 5 1q E
5 Unqq Uny 5 E 5 Un 5 E
qc
5 Unqq Uny 5 E a 5 Un 5 1q E
ª0
as n ª `.
Therefore, by Lemma 2.1Ži., we have lim inf nª`
HV
=F˜Ž X , Un . ? Un0 a 5 Un 5 1q E
s lim inf nª`
HV
=F˜Ž X , Un . ? Un a 5 Un 5 1q E
) 0,
which contradicts the upper limit above. The proofs of the claim and hence of the lemma are complete. Next, we use the above two lemmas to prove Theorems 1.1 and 1.2. Proof of Theorem 1.1. Set E1 s N [ V, E2 s W. We first claim that I ŽU . ª q` for U g E2 as 5 U 5 E ª ` and that I ŽU . ª y` for U g E1 as 5 U 5 E ª `. Indeed, since ŽZ 1 . implies that < F˜Ž X, U .< F cŽ< U < 1q a q 1. for all U g R 2 and for some positive constant c and noting that I < E 2 is positive, then there exists m ) 0 such that I Ž U . s 12 ² AU, U :E y
HVF˜Ž X , U . G
1 2
a m 5 U 5 2E y c 5 U 5 1q ycª` E
as U g E2 and 5 U 5 E ª `. On the other hand, there exist a constant M0 and a sequence Un s Un0 q Uny g E1 s N [ V with 5 Un 5 E ª ` such that I ŽUn . G M0 , which implies that lim inf nª`
I Ž Un . a 5 Un 5 1q E
G0
and
lim inf nª`
I Ž Un . 5 Un 5 2E
G 0.
Therefore, if Uny r5 Un 5 E ª 0 as n ª `, we have, by Lemma 2.1, lim inf nª`
HV
F˜Ž X , Un . a 5 Un 5 1q E
) 0.
408
WENMING ZOU
Hence, I ŽUn . s 12 ² AUn , Un :E y HV F˜Ž X, Un . F yHV F˜Ž X, Un . implies that lim sup nª`
I Ž Un . a 5 Un 5 1q E
- 0,
which contradicts the preceding estimation about the lower limit. On the contrary, if Uny r5 Un 5 E ¢ 0, then there exists a subsequence, say Un4 , such that lim inf nª` 5 Uny5 Er5 Un 5 E ) 0. Since I Ž Un . s 12 ² AUny, Uny :E y
HVF˜Ž X , U . n
a F y 12 c 5 Uny 5 2E q c 5 Un 5 1q q c, E
it follows that
lim sup nª`
I Ž Un . 5 Un 5 2E
5 Uny5 2E 1 F y c lim inf - 0, 2 nª` 5 Un 5 2E
which also contradicts the preceding lower limit. The above arguments show that our claim is true, and consequently, Theorem 1.1 follows from Lemma 2.2 and the Saddle Point Theorem Žcf. w10x.. Proof of Theorem 1.2. First, by condition ŽZ 3 ., it is easy to check that
d 1 F inf
U/0
2 F˜Ž X , U .
2
and
d 2 G lim inf Uª0
2 F˜Ž X , U .
.
Moreover, for any « 0 ) 0, « 0 q d 2 - 0, there exists r 1 ) 0 such that F˜Ž X,U . F 12 Ž d 2 q « 0 .< U < 2 for a.e. X g V and for < U < F r 1. Now, combin˜ we have two constants C ) 0 ing condition ŽZ 1 . and the definition of F, and p ) 2 such that 1 2
d 1 < U < 2 F F˜Ž X , U . F 21 Ž d 2 q « 0 . < U < 2 q C < U < p .
Setting E3 s Ž Hlyky 1 [ Hl0ky 1 . = Ž Hlyj [ Hl0j ., E4 s E3H s Hlqky 1 = Hlqj , then 1 - dim E3 - q` Žsince k G 2.. Now to prove Theorem 1.2, we have only to check the geometry condition of the Generalized Mountain Pass Theorem Žcf. w10x.. For any
409
SOLUTIONS FOR RESONANT ELLIPTIC SYSTEMS
U s Ž u, ¨ . g E4 s Hlqky 1 = Hlqj , we have IŽU . s G
1 2 1 2
² AU, U :E y 5 u5 2 y
y G
1 2
1 2
1 2
HVF˜Ž X , U .
l k 5 u 5 22 q
1 2
5¨ 52 y
1 2
z 5 ¨ 5 22 Ž d 2 q « 0 . 5 u 5 22
Ž d 2 q « 0 . 5 ¨ 5 22 y c 5 U 5 Ep
5 u5 2 1 y
ž
lk q d 2 q « 0 lk
/
q
1 2
5¨ 52 1y
ž
z q d2 q «0 lj
/
y c 5 U 5 Ep
G c 5 U 5 2E y c 5 U 5 Ep . Hence, there exist a ) 0, r ) 0 such that I ŽU . G a ) 0 for U g E4 with 5 U 5 E s r . On the other hand, for any U s Ž u, ¨ . g E3 , we have I Ž U . s 12 ² AU, U :E y
HVF˜Ž X , U .
F 12 5 u 5 2 y 12 l k 5 u 5 22 q 21 5 ¨ 5 2 y 21 z 5 ¨ 5 22 y 21 d 1 5 u 5 22 y 21 d 1 5 ¨ 5 22 F 0. Furthermore, setting E5 s Ž Hlyk [ Hl0k . = Ž Hlyj [ Hl0j ., then E3 ; E5 ; E1 s V [ N
and
dim E5 y dim E3 G 1,
and, by the claim in the proof of Theorem 1.1, I ŽU . ª y` for U g E5 as 5 U 5 E ª `. Now combining the above geometry condition, Lemma 2.2, and the Generalized Mountain Pass Theorem Žcf. Theorem 5.3 and Remark 5.5 of w10x., we get at least one nontrivial critical point with a positive critical value. 3. =F IS ODD To prove Theorems 1.3 and 1.4, we have to establish some lemmas. LEMMA 3.1. Assume ŽO 2 .. For any l m g s ŽyD . with l m 4 max j , z 4 , y. 0 . Ž y Ž 0 Ž . set Ey m s Hl m = Hl m [ Hl m = Hl m . Then there exist r s r m ) 0, b s b Ž m. - 0 such that IŽU . F b - 0
5 5 for U g Ey m l U g E, U E s r 4 .
Furthermore, lim m ª` r Ž m. s 0.
410
WENMING ZOU
Proof. Since condition ŽO 2 . implies that lim inf
2 F˜Ž X , U .
Uª0
s q`
uniformly for a.e. X g V ,
then, for l m , there exists r s r Ž m. ) 0 such that 2 F˜Ž X , U .
G 2 max l m y j , l m y z 4
for a.e. X g V and for < U < F r .
Setting F˜1Ž X, U . s F˜Ž X, U . y 12 Ž l m y j .< u < 2 y 12 Ž l m y z . ¨ 2 for a.e. X g V and for 0 - < U < 2 s u 2 q ¨ 2 F r 2 , then F1Ž X, U . ) 0. Since Ž . < < < Ž .< dim Ey m - `, then there exists C 0 s C 0 m such that U F sup U X : y 4 Ž .5 5 X g V F C0 m U E for all U g Em . If we choose r s r Ž m. F 5 U 5 E s r Ž m.4 implies that r Ž m.rŽ C0 Ž m. q 1., then U g Ey m l U g E: < UŽ X .< F r Ž m., hence I Ž U . s 12 ² AU, U :E y
HV Ž
1 2
Ž l m y j . u 2 q 12 Ž l m y z . ¨ 2 q F˜1 Ž X , U . .
HVF˜ Ž X , U . - 0.
Fy
1
5 5 Ž .4 Ž . Since Ey m l U g E: U E s r m is compact, there exists b s b m - 0 5 5 4 such that I ŽU . F b - 0 for U g Ey m l U g E, U E s r . Finally, from the above arguments, it is evident that lim m ª` r Ž m. s 0. LEMMA 3.2. Assume ŽO 3 ., U s Uqq U 0 q Uyg E s W [ N [ V. Then, for any constant D 1 ) 0, we ha¨ e that lim 5U 0 5 E ª` J9ŽU . s 0 uniformly for 5 Uqq Uy 5 E F D 1. Proof. Condition ŽO 3 . implies that =F˜Ž X, U . ª 0 as < U < ª `. Hence 1r2
² J9 Ž U . , f: E s H =F˜Ž X , U . ? F F c 5 F 5 E V
ž
HV =F˜Ž X , U .
/
.
Since N s Ker A has the unique continuation property and E is compactly embedded in L p = L p for p - 2*, using arguments similar to those in the proof of Lemma 3.2 of w5x, we have
HV =F˜Ž X , U .
ª0
as 5 U 0 5 E ª ` uniformly for 5 Uqq Uy 5 E F D 1 ,
that is, lim 5U 0 5 E ª` J9ŽU . s 0 uniformly for 5 Uqq Uy 5 E F D 1.
411
SOLUTIONS FOR RESONANT ELLIPTIC SYSTEMS
LEMMA 3.3. that
Assume ŽOX3 .. Then, for any D ) 0, there exists L ) 0 such
² I9 Ž U . , U: E q 12 I9 Ž U .
E
? 5U 5 E G 0
for U g SŽ D, L. [ U s Uqq Uyq U 0 g E s W [ V [ N: I ŽU . G yD, 5 Uqq Uy 5 E G L4 . Proof. Condition ŽOX3 . implies that < =F˜Ž X, U .< F 2 M0 for a.e. X g V and U g R 2 . Hence 5 J X ŽU .5 E F c for any U g E and for some positive constant c. Now for any U g SŽ D, L., where the constant L will be determined later, we have
² I9 Ž U . , U: E q
1
I9 Ž U . ? 5 U 5 E
2
s 2 IŽU . q J ŽU . y
ž
q
1 2
G 2 5U 5 E G G
ž ž
1 4 1 4
1 2
² J9 Ž U . , U: E
A Ž Uqq Uy . y J9 Ž U .
ž
1
A Ž Uqq Uy .
4
c 5 Uqq Uy 5 E y cL y
7 4
cy
D L
/
7 4
E
E
y
? 5U 5 E
7 4
D
cy
/
5U 5 E
cy
/
D 5U 5 E
/
5U 5 E
L G 0,
if we choose L appropriately large. Now, to prove Theorems 1.3 and 1.4, we use a new and interesting theorem due to Fei w3x. Before stating it, we first recall the definition of condition ŽB.. Let E be a real Hilbert space with norm 5 ? 5 E and inner product ² ? , ? :E . We say that I g C 1 Ž E, R. satisfies condition ŽB. in Ž c1 , c 2 . ; wy`, q`x relative to the ball B [ U g E, 5 U 5 E F r 4 for some r ) 0 if Ži. any sequence Un4 ; B such that I ŽUn . ª c g Ž c1 , c 2 . and I9ŽUn . ª 0 is such that Un4 possesses a convergent subsequence; and Žii. for any c g Ž c1 , c 2 ., there exist s s s Ž c . g Ž0, min c y c1 , c 2 y c4. and u - 1 such that
² I9 Ž U . , U: E q u I9 Ž U . for any U g Iy1 Žw c y s , c q s x. l B.
E
? 5U 5 E G 0
412
WENMING ZOU
Remark 3.1. For system ŽP., since =F Ž =F˜. satisfies the subcritical growth condition, the corresponding functional I automatically satisfies condition ŽB.Ži.. THEOREM 3.1.
Suppose that I g C 1 Ž E, R. is e¨ en and satisfies
Ži. There exist closed subspaces E1 and E2 of E with codim E2 - q` and two positi¨ e constants r ) 0, b ) y` such that IŽU . - 0
sup UgE1lSr
inf I Ž U . ) b,
and
UgE 2
where Sr s U g E: 5 U 5 E s r 4 . Žii. I satisfies condition ŽB. in Ž b, 0. relati¨ e to B s U g E: 5 U 5 E F r, r ) r 4 . Then if dim E1 y codim E2 ) 0, I has at least dim E1 y codim E2 distinct pairs of critical points, whose corresponding critical ¨ alues belong to Ž b, 0.. Now we are ready to prove the existence of infinitely many solutions in the case that =F is odd. Proof of Theorem 1.3. We first prove the following claim: lim sup 05
5 U Eª` U 0 gN
HVF˜Ž X , U
0
. F 0.
Condition ŽO4 . implies that F˜Ž X, U . F 0 for < U < G R 0 . Moreover, since N has the unique continuation property, we have that Žcf. Lemma 3.2 in w5x., for any « ) 0, there exists d Ž « . ) 0 such that mes X g V : U 0 Ž X . - d Ž « . 5 U 0 5 E 4 - «
for any U 0 g N _ 0 4 .
Letting V ŽU 0 . [ X g V: < U 0 Ž X .< G d Ž « .5 U 0 5 E 4 and choosing R1 with R1 G R 0rd Ž « ., then for 5 U 0 5 E G R1 and X g V ŽU 0 . we have U 0 Ž X . G R0
and hence
F˜Ž X , U 0 . F 0.
Setting V1 s X g V : U 0 Ž X . ) R0 4 l Ž V _ V Ž U 0 . . , V 2 s X g V : U 0 Ž X . F R0 4 l Ž V _ V Ž U 0 . . , then mes V 2 - « and
HV_V ŽU .F˜Ž X , U 0
0
. F H F˜Ž X , U 0 . F « max V2
XgV < U 0
F˜Ž X , U 0 . s c « .
413
SOLUTIONS FOR RESONANT ELLIPTIC SYSTEMS
It follows that, for 5 U 0 5 E G R1 ,
HV F˜Ž X , U
0
. sH
V ŽU 0 .
F˜Ž X , U 0 . q
HV_V ŽU .F˜Ž X , U 0
0
. F « c,
that is, lim sup
H F˜Ž X , U
5 U 0 5 Eª` V U 0 gN
0
. F 0.
Now, condition ŽO 3 . implies that there exists M0 ) 0 such that < =F Ž X, U .< F M0 and hence < =F˜Ž X, U .< F M0 for a.e. X g V and for all U g R 2 . It follows that < F Ž X, U .< F '2 M0 < U < and < F˜Ž X, U .< F '2 M0 < U < for a.e. X g V and for all U g R 2 . In particular, 5 J9ŽU .5 E F c for all U g E and for some constant c ) 0. Next, for any « g Žy1, 0., we choose l« g s ŽyD . with l« ) max 2 j , 2 z , 4 M02 mes VrŽy« .4 . We now fix l« and set E2 s Hlq« = Hlq« , then codim E2 - ` and for any U s Ž u, ¨ . g E2 , IŽU . s G G
1 2 1 2 1 4
² AU, U :E y
ž
1y
j l«
/
HVF˜Ž X , U .
5 u5 2 q
1 2
ž
1y
z l«
/
5 ¨ 5 2 y '2 M0
5 U 5 2E y 2 M0 Ž mes V . 1r2 5 U 5 Erl1r2 Gy «
HV
4 M02 mes V
l«
)«,
that is, inf U g E 2 I ŽU . ) « . Finally, we prove that I satisfies condition ŽB.Žii. in Ž « , 0. relative to the set B s U g E: 5 U 5 E F r 4 , where r will be determined later. Indeed, for any d g Ž « , 0., letting s s 14 min yd, d y « 4 ) 0, we obtain L [ w d y s , d q s x ; Ž « , 0.. If U g Iy1 Ž L ., i.e., I ŽU . G d y s G y< « <, then by Lemma 3.3, there exists L« ) 0 such that
² I9 Ž U . , U: E q 12 I9 Ž U .
E
? 5U 5 E G 0
for U g SŽ< « <, L« . [ U s Uqq U 0 q Uyg E s W [ N [ V: I ŽU . G y< « <, 5 Uqq Uy 5 E G L« 4 . If U g Iy1 Ž L . and 5 Uqq Uy 5 E F L« , then we have, by Lemma 3.2, that lim 5U 0 5 E ª` J9ŽU . s 0 uniformly for 5 Uqq Uy 5 E F L« . Therefore, for any « 1 small enough, there exists L1 ) 0 such that < J ŽU . y J ŽU 0 .< - « for 5 U 0 5 E G L1. By the claim at the beginning in this proof, we know that
414
WENMING ZOU
there exists L2 ) 0 such that J ŽU 0 . - « 1 for 5 U 0 5 E G L2 . Consequently, we obtain that 1 2
² AU, U :E s I Ž U . q J Ž U . F d q s q « 1 q « 1 - 0
for U g Iy1 Ž L . with 5 Uqq Uy 5 E F L« and 5 U 0 5 E G L1 q L2 . It follows that A Ž Uqq Uy .
E
y2 Ž d q s q 2 « 1 .
G
L«
) 0.
Lemma 3.2 implies that there exists L3 4 L1 q L2 such that for any U g Iy1 Ž L . with 5 Uqq Uy 5 E F L« , 5 U 0 5 E G L3 , we have
² I9 Ž U . , U: E q
1 2
I9 Ž U .
E
s 2 IŽU . q J ŽU . y
ž
q
1
G2
ž
1 2
² J9 Ž U . , U: E
A Ž Uqq U 0 . y J9 Ž U .
4
G 2 5U 5 E
? 5U 5 E
ž
1 4
A Ž Uqq Uy .
yŽ d q s q 2 « . 2 L«
y
E
3 4
y
E
3 4
J9 Ž U .
? 5U 5 E J9 Ž U .
E
y
/ E
q
J ŽU . q IŽU .
2«1 q « 5U 5 E
5U 5 E
/
/
5 U 5 E G 0.
Combining the above arguments, we obtain that I satisfies condition ŽB.Žii. in Ž « , 0. relative to B s U g E, 5 U 5 E F r, r G L1 q L2 q L3 q L« q 14 . Hence, Remark 3.1, Lemma 3.1, and Theorem 3.1 with E1 s Ey m imply that The number of critical points of I G dim E1 y codim E2 s dim Ž Hlym = Hlym . m Ž Hl0m = Hl0m .
ž
/
y dim Ž Hly« = Hly« . [ Ž Hl0« = Hl0« . ,
ž
/
and the corresponding critical values are in Ž « , 0.. Therefore, keeping in mind Lemma 3.1 and passing to the limit as m ª `, we obtain infinitely many critical points of I, the critical values of which belong to Ž « , 0.. Finally, since « g Žy1, 0. is arbitrary, we have a critical point sequence Un4`ns1 such that I ŽUn . ª 0y as n ª `, as desired.
SOLUTIONS FOR RESONANT ELLIPTIC SYSTEMS
415
Proof of Theorem 1.4. Conditions ŽO 1 ., ŽO 2 ., ŽOX3 ., and ŽOX4 . hold. It is easy to verify that condition ŽOX4 . implies that F˜Ž X , U . y 12 =F˜Ž X , U . ? U G h Ž X . for a.e. X g V and for < U < G R 0 . First, by condition ŽOX4 ., we prove that, for any L ) 0, U s Uqq U 0 q U g W [ N [ V s E implies that y
H Ž F˜Ž X , U . y
lim
5 U 0 5 Eª` V 5 U qqU y 5FL
1 2
=F˜Ž X , U . ? U . G
HVh Ž X . ) 0.
To prove this, we again use an argument similar to that in w5x. For any « 1 ) 0 and « 2 ) 0, there exist d Ž « 1 . ) 0 and d Ž « 2 . ) 0 such that mes X g V : U 0 Ž X . - d Ž « 1 . 5 U 0 5 E 4 - « 1
for any U 0 g N _ 0 4 ,
mes X g V : Uq Ž X . q Uy Ž X . ) d Ž « 2 . 5 Uqq Uy 5 E 4 - « 2 for any Uqq Uyg W [ V . We choose C0 s Ž R 0 q d Ž « 2 . L.rd Ž « 1 ., where R 0 is in ŽOX4 ., and let U be such that 5 U 0 5 E ) C0 . Setting V Ž U 0 . s X g V : U 0 Ž X . G d Ž « 1 . 5U 0 5 E 4 , V Ž Uqq Uy . s X g V : Uq Ž X . q Uy Ž X . F d Ž « 2 . 5 Uqq Uy 5 E 4 , then, mesŽ V _ V ŽU 0 .. - « 1 , mesŽ V _ V ŽUqq Uy .. - « 2 . Moreover, if X g V ŽU 0 . l V ŽUqq Uy ., then U Ž X . G U 0 Ž X . y Uq Ž X . q Uy Ž X . G d Ž « 1 . 5 U 0 5 E y d Ž « 2 . 5 Uqq Uy 5 E G C0 d Ž « 1 . y d Ž « 2 . L s R 0 . It follows that F˜Ž X, U . y 12 =F˜Ž X, U . ? U G hŽ X .. If X g V _ Ž V ŽU 0 . l V ŽUqq Uy .. [ V 1 , then mes V 1 F « 1 q « 2 , and hence
HV Ž F˜Ž X , U . y
1 2
=F˜Ž X , U . ? U y h Ž X . .
1
G
HV
1
< U
Ž F˜Ž X , U . y
1 2
=F˜Ž X , U . ? U y h Ž X . . G yc1 Ž « 1 q « 2 . ,
416
WENMING ZOU
where c1 s max F˜Ž X, U . y 12 =F˜Ž X, U . ? U y hŽ X .: X g V, < U < F R 0 4 . Combining the above two cases, for 5 U 0 5 E G C0 , we have
HV Ž F˜Ž X , U . y
1 2
=F˜Ž X , U . ? U y h Ž X . . G yc1 Ž « 1 q « 2 . .
It follows that lim 5U 0 5 E ª` HV Ž F˜Ž X, U . y 12 =F˜Ž X, U . ? U . G HV hŽ X . ) 0, since « 1 and « 2 are given arbitrarily. Second, note that conditions ŽOX3 . and ŽO 1 . imply that < F˜Ž X, U .< F '2 M0 < U < for any U g R 2 and for a.e. X g V Žhere M0 comes from ŽOX3 ... Therefore, for any « g ŽyCh , 0., where Ch s HV hŽ X . ) 0, we take l« g s ŽyD . with l« ) max 2 j , 2 z , 4 M02 mes VrŽy« .4 and define E2 s Hlq« = Hlq« . Then codim E2 - ` and, for any U s Ž u, ¨ . g E2 , by using the same calculation as that in the proof of Theorem 1.3, we have that inf U g E 2 I ŽU . ) «. Next we show that the functional I satisfies condition ŽB.Žii. in Ž « , 0. relative to B [ U g E: 5 U 5 E F r 4 , where r is to be determined later. Taking e g Ž « , 0. and letting s s 14 min e y « , < e <4 , we have that L [ w e y s , e q s x ; Ž « , 0.. Now, for U g Iy1 Ž L ., there exists L« ) 0, by Lemma 3.3, such that
² I9 Ž U . , U: E q 12 I9 Ž U .
E
? 5U 5 E G 0
if 5 Uqq Uy 5 E G L« .
But if 5 Uqq Uy 5 F L« , we have
² I9 Ž U . , U: E q 12 I9 Ž U .
E
G 2Ž I Ž U . q J Ž U . y
? 5U 5 E 1 2
² J9 Ž U . , U: E .
HVF˜Ž X , U . y
G 2Ž e y s . q 2
1 2
=F˜Ž X , U . ? U.
The first part of this proof implies that lim
H F˜Ž X , U . y
5 U 0 5 Eª` V 5 U qqU y 5 EFL«
1 2
=F˜Ž X , U . ? U G Ch ) 0.
Therefore, for « 1 ) 0 small enough, there exists L1 such that, for 5 U 0 5 E G L1 , we have
² I9 Ž U . , U: E q 12 I9 Ž U .
E
? 5U 5 E
G 2 Ž e y s q Ch y « 1 . G 2 Ž « q Ch y « 1 . G 0. Summing up, if we choose r G L1 q L« , then I satisfies condition ŽB.Žii. in Ž « , 0. relative to B s U g E: 5 U 5 E F r 4 . Consequently, combining Lemma
SOLUTIONS FOR RESONANT ELLIPTIC SYSTEMS
417
y 3.1, Remark 3.1, and Theorem 3.1 with E1 s Em , we have that the number of the critical points of I G dim E1 y codim E2 . As before, we get again a critical point sequence Un4`ns0 such that I ŽUn . ª 0y as n ª `.
Proof of Corollary 1.1. The proof follows from Theorems 1.3 and 1.4.
ACKNOWLEDGMENTS I want to take the opportunity to thank Prof. Shujie Li for advice and a lot of helpful discussion. I also thank the referee for careful reading of the paper.
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