Multiplicity of small negative-energy solutions for a class of nonlinear Schrödinger–Poisson systems

Applied Mathematics and Computation 243 (2014) 817–824

Contents lists available at ScienceDirect

Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc

Multiplicity of small negative-energy solutions for a class of nonlinear Schrödinger–Poisson systems q Liping Xu a,b, Haibo Chen a,⇑ a b

School of Mathematics and Statistics, Central South University, Changsha, 410075 Hunan, People’s Republic of China Department of Mathematics and Statistics, Henan University of Science and Technology, Luoyang 471003, People’s Republic of China

a r t i c l e

i n f o

Keywords: Schrödinger–Poisson system Variant fountain theorem Variational approaches

a b s t r a c t This paper deals with the following nonlinear Schrödinger–Poisson systems

(

Du þ VðxÞu þ KðxÞ/ðxÞu ¼ HðxÞf ðx; uÞ; in R3 ; D/ ¼ KðxÞu2 ; in R3 ;

where VðxÞ, KðxÞ and HðxÞ are nonnegative continuous functions. Under appropriate assumptions on VðxÞ, KðxÞ; HðxÞ and f ðx; uÞ, we prove the existence of infinitely many small negative-energy solutions by using the variant fountain theorem established by Zou. Recent results from the literature are extended. Ó 2014 Published by Elsevier Inc.

1. Introduction and main results In this paper, we are interested in the existence and multiple solutions for the following Schrödinger–Poisson system:

(

Du þ VðxÞu þ KðxÞ/ðxÞu ¼ HðxÞf ðx; uÞ; in R3 ; D/ ¼ KðxÞu2 ; in R3 ;

ð1:1Þ

where the functions VðxÞ; KðxÞ and HðxÞ are nonnegative continuous. We assume that the functions VðxÞ; KðxÞ; HðxÞ and f ðx; uÞ satisfy the following hypotheses. (H1) VðxÞ 2 CðR3 ; RÞ satisfies inf x2R3 VðxÞ P a > 0, where a > 0 is a constant. Moreover, for any M > 0; measfx 2 R3 : VðxÞ 6 Mg < 1, where measð:Þ denotes the Lebesgue measure in R3 . (H2) KðxÞ 2 L1 ðR3 ; RÞ, and KðxÞ P 0 for any x 2 R3 . T 2 (H3) HðxÞ 2 L2m ðR3 ; RÞ L1 ðR3 ; RÞ; HðxÞ P 0; f 2 CðR3 ; RÞ and there exists constant b > 0 such that jf ðx; uÞj 6 bðjuj þ jujm1 Þ, where 1 < m < 2. (H4) Fðx; 0Þ ¼ 0; Fðx; uÞ P 0 for all ðx; uÞ 2 R3  R, and for some 1 < l < 2, there exists constant c1 > 0 such that Ru Fðx; uÞ P c1 jujl ; 8ðx; uÞ 2 R3  R, where Fðx; uÞ :¼ 0 f ðx; sÞds. 3 (H5) f ðx; uÞ ¼ f ðx; uÞ; 8ðx; uÞ 2 R  R.

q

Research supported by Natural Science Foundation of China 11271372 and by Hunan Provincial Natural Science Foundation of China 12JJ2004.

⇑ Corresponding author.

E-mail addresses: [email protected] (L. Xu), [email protected] (H. Chen). http://dx.doi.org/10.1016/j.amc.2014.06.043 0096-3003/Ó 2014 Published by Elsevier Inc.

818

L. Xu, H. Chen / Applied Mathematics and Computation 243 (2014) 817–824

System (1.1) is also known as the nonlinear Schrödinger–Maxwell system, which arises in an interesting physical context. If we look for solitary solutions of Schrödinger equation for a particle in an electrostatic field, we just need to solve (1.1). We refer the interest readers to [1,2] for more details on the physical aspects. With the aid of variational methods, system (1.1) has been extensively investigated in the literature under various hypotheses on the potential VðxÞ; KðxÞ and the nonlinear term over the past several decades. See, e.g., [1–19] and the references quoted in them. Most of the literatures focus on the study of existence, nonexistence of solutions, multiplicity of solutions, ground states, radially and non-radial solutions of such a system with f ðx; uÞ ¼ jujp1 u (see [3–10]). For example, in [3,4], the authors proved the existence of infinitely many pairs of high energy radial solutions with 2 < p < 5, and also proved some existence results for 1 < p < 2 and p ¼ 2. In [6], Sun studied the existence of infinitely many solutions for some p 2 ð0; 1Þ. Giovanna and Giusi [7] proved the existence of positive solutions for 3 < p < 5 under the case of the lack of compactness. Moreover, there have been many papers dealing with the existence and multiplicity of solutions for systems (1.1) with VðxÞ or KðxÞ being a constant, or both of them being constants (see [3,6–16]). In [11], the authors studied the existence of infinitely many large energy solutions for the superlinear Schrödinger–Maxwell equations with KðxÞ ¼ 1 via the Fountain Theorem in critical point theory. The authors [14] proved the existence and multiplicity of semiclassical solutions for Schrödinger–Maxwell system for KðxÞ ¼ 1 by variational methods, too. Azzollini et al. [16] studied the existence of a nontrivial solution to the nonlinear Schrödinger–Maxwell equations for VðxÞ ¼ 0; KðxÞ ¼ constant in R3 . The natural question is to consider Schrödinger–Poisson system (1.1) in R3 , where f ðx; uÞ – jujp1 u, VðxÞ – constant and KðxÞ – constant. For this case, it is worth noticing that there are few works concerning on the existence of infinitely many small negative-energy solutions for system (1.1). Motivated by the above fact, in this paper, we give our positive answer. Our tool is the variant fountain theorem established in [17]. Our main results are the following: Theorem 1.1. Assume that (H1)–(H5) hold, then problem (1.1) possesses infinitely many solutions fðuk ; /k Þg satisfying

1 2

Z

R3

ðjruk j2 þ VðxÞu2k Þdx 

1 4

Z

R3

jr/k j2 dx þ

1 2

Z

R3

KðxÞ/k u2k dx 

Z

R3

HðxÞFðx; uk Þdx ! 0 as k ! 1:

Remark 1.1. (i) The condition (H1) originally introduced in [19] is used to obtain the compact embedding theorem of the working space E. (ii) The author [6] obtained infinitely many small negative-energy solutions for (1.1) with KðxÞ ¼ 1;f ðx;uÞ ¼ ðp þ 1ÞbðxÞjujp1 u; 0 < p < 1. For this cause, we go on step further. The rest of the paper is given as follows: in Section 2, we present some preliminary results, which are necessary to Sections 3. In Section 3, we give the proof of Theorem 1.1. Throughout the paper we shall denote by C > 0 various positive constants. 2. Preliminaries We first recall the following notations. Define the function space

E ¼ fu 2 H1 ðR3 Þ :

Z R3

ðjruj2 þ VðxÞu2 Þdx < þ1g;

then E is a Hilbert space equipped with the inner product and norm

ðu; v Þ ¼

Z R3

kuk ¼ ðu; uÞ1=2 :

ðru  rv þ VðxÞuv Þdx;

Since VðxÞ is bounded from below, E is continuously embedded into Lq ðR3 Þ for all q 2 ½2; 2 , where 2 ¼ 6 is the critical exponent for the Sobolev embedding in dimension 3. Therefore, there exists a positive constant C > 0 such that

kukLq 6 Ckuk; where kukLq :¼ ð

R

R3

8u 2 E; q

ð2:1Þ

1 q

q

3

juj dxÞ for q 2 ½1; 1Þ is the norm of the usual Lebesgue space L ðR Þ. Moreover, by (H1), the embedding

3 E,!Lq ðR3 Þ is also compact for any q 2 ½2; 2 Þðcf :½20Þ. Define the function space D1;2 ðR3 Þ is the completion of C 1 0 ðR Þ with 1 R respect to the Dirichlet norm kukD1;2 ¼ ð R3 jruj2 dxÞ2 . It is well known that the embedding D1;2 ðR3 Þ,!L6 ðR3 Þ is continuous (cf. [21]). It is clear that system (1.1) is the Euler–Lagrange equations of the functional J : E  D1;2 ðR3 Þ ! R defined by

Jðu; /Þ ¼

1 1 kuk2  2 4

Z R3

jr/j2 dx þ

1 2

Z R3

KðxÞ/u2 dx 

Z R3

HðxÞFðx; uÞdx:

819

L. Xu, H. Chen / Applied Mathematics and Computation 243 (2014) 817–824

Obviously, the functional J belongs to C 1 ðE  D1;2 ðR3 Þ; RÞ and its critical points are the solutions of (1.1). It is easy to know that J exhibits a strong indefiniteness. However this indefiniteness can be removed using the reduction method described in [22], by which we are led to study a variable functional that does not present such a strongly indefinite nature. Now, we recall this method. R For any u 2 E, consider the linear functional T u : D1;2 ðR3 Þ ! R defined as T u ðv Þ ¼ R3 KðxÞu2 v dx. Since the following embed

ding E,!Lq ðR3 Þ; 2 6 q 6 2 ; D1;2 ðR3 Þ,!L2 ðR3 Þ are continuous and KðxÞ 2 L1 ðR3 Þ; KðxÞ P 0, we have

Z R3

Ku2 v dx 6 kKk1 kuk2L12=5 kv kL6 6 Ckuk2 kv kD1;2 :

ð2:2Þ

So, T u is continuous on D1;2 ðR3 Þ. It follows from the Lax–Milgram theorem that for every u 2 E, there exists a unique /u 2 D1;2 ðR3 Þ such that

M/u ¼ KðxÞu2 ;

ð2:3Þ

and /u can be represented by

/u ¼

1 4p

Z

KðyÞu2 ðyÞ dy; jx  yj

R3

3 C1 0 ðR Þ

for any u 2 (see [23], Theorem 1), by density it can be extended for any u 2 E (also, see Lemma 2.1 of [24]). It follows from KðxÞ P 0 that /u P 0, and /u ¼ /u for any u 2 E. By the Hölder inequality, the Sobolev inequality and (2.3), for any u 2 E, it is easy to see that, if KðxÞ 2 L1 ðR3 Þ,

k/u k2D1;2 ¼

Z R3

KðxÞu2 /u dx 6 kKk1 kuk2L12=5 k/u k6 6 Ckuk2L12=5 k/u kD1;2 ;

then it follows that

k/u kD1;2 6 Ckuk2L12=5 :

ð2:4Þ

Hence, by the Hölder inequality, (2.2) and (2.4), we have

Z R3

KðxÞ/u u2 dx 6 Ckuk4L12=5 6 Ckuk4 :

ð2:5Þ

So, we can consider the functional I : E ! R defined by IðuÞ ¼ Jðu; /u Þ. By (2.3), the reduced functional takes the form

IðuÞ ¼

1 1 kuk2 þ 2 4

Z

R3

KðxÞ/u u2 dx 

Z

HðxÞFðx; uÞdx:

ð2:6Þ

R3

By (H3), (H4) and Hölder inequality, we easily see that

06

Z

HðxÞFðx; uÞdx 6

R3

Z R3

CHðxÞðjuj2 þ jujm Þdx 6 CðkHk1 kuk2L2 þ kHk

2

L2m

kukmL2 Þ 6 Cðkuk2 þ kukm Þ:

ð2:7Þ

It follows from (2.5) and (2.7) that I is well defined. Furthermore, I is a C 1 functional with derivative given by

ðI0 ðuÞ; v Þ ¼

Z R3

ðru  rv þ VðxÞuv þ KðxÞ/u uv  HðxÞf ðx; uÞv Þdx:

Now, we can apply Theorem 2.3 of [22] to our functional I and obtain: Lemma 2.1. The following statements are equivalent: (1) ðu; /Þ 2 E  D1;2 ðR3 Þ is a critical point of J (i.e. ðu; /Þ 2 E is a solution of (1.1)); )(2) u is a critical point of I and / ¼ /u . To complete the proof of our theorem, the following theorem will be needed in our argument. Let E be a Banach space k

1

with k:k and E ¼ aj2N X j with dimX j < 1 for any j 2 N. Set Y k ¼ aj¼0 X j ; Z k ¼ aj¼kþ1 X j . Consider the following C 1 functional wk : E ! R defined by

wk ðuÞ ¼ AðuÞ  kBðuÞ;

k 2 ½1; 2;

where A; B : E ! R are two functionals. Theorem 2.1 (See [17, Theorem 2.2]). Suppose that the functional wk ðuÞ defined above satisfies: (F1) wk maps bounded sets to bounded sets uniformly for k 2 ½1; 2. Furthermore, wk ðuÞ ¼ wk ðuÞ for all ðk; uÞ 2 ½1; 2  E. (F2) BðuÞ P 0; BðuÞ ! 1 as kuk ! 1 on any finite dimensional subspace of E.

820

L. Xu, H. Chen / Applied Mathematics and Computation 243 (2014) 817–824

(F3) There exist qk > r k > 0 such that

ak ðkÞ :¼ inf u2Zk ;kuk¼qk wk ðuÞ P 0 > bk ðkÞ :¼ maxu2Y k ;kuk¼rk wk ðuÞ; k 2 ½1; 2: dk ðkÞ :¼ inf u2Zk ;kuk6qk wk ðuÞ ! 0 as k ! 1 uniformly for k 2 ½1; 2, Then there exist kn ! 1; uðkn Þ 2 Y n such that

w0kn jY n ðuðkn ÞÞ ¼ 0; wkn ðuðkn ÞÞ ! ck 2 ½dk ð2Þ; bk ð1Þ as n ! 1: In particular, if fuðkn Þg has a convergent subsequence for every k, then w1 has infinitely many nontrivial critical points fuk g  E n f0g satisfying w1 ðuk Þ ! 0 as n ! 1. 3. Proof of the main result Next, we apply Theorem 2.1 to prove our main result. So, we define the functionals A; B and wk on our working space E by

AðuÞ ¼

1 1 kuk2 þ 2 4

Z

R3

KðxÞ/u u2 dx;

and

wk ðuÞ ¼ AðuÞ  kBðuÞ ¼

1 1 kuk2 þ 2 4

BðuÞ ¼ Z R3

Z

HðxÞFðx; uÞdx;

ð3:1Þ

R3

KðxÞ/u u2 dx  k

Z

HðxÞFðx; uÞdx;

ð3:2Þ

R3

for all u 2 E and k 2 ½1; 2. Evidently, wk 2 C 1 ðE; RÞ for all k 2 ½1; 2. We choose a completely orthonormal basis fej : j 2 Ng of E and let X j ¼ spanfej g for all j 2 N. Note that w1 ¼ I, where I is the functional defined in (2.6). To complete the proof of our theorem, we need the following lemmas. Lemma 3.1. Let (H1)–(H4) be satisfied, then BðuÞ P 0. Furthermore, BðuÞ ! 1 as kuk ! 1 on any finite dimensional subspace of E. Proof. Obviously, by (H4), BðuÞ ¼ exists  > 0 such that

R

R3

HðxÞFðx; uÞdx P 0. Now we claim that for any finite dimensional subspace F  E, there

measfx 2 R3 : HðxÞjuðxÞjl P kukl g P ;

8u 2 F n f0g:

ð3:3Þ

Arguing by contradiction, we assume that there exists a sequence fun ðxÞgn2N  F n f0g such that

  1 1 meas x 2 R3 : HðxÞjun ðxÞjl P kun kl < ; n n

Let

8n 2 N:

v n ¼ kuu k  F n f0g, Then kv n k ¼ 1 for all n 2 N, and n n

  1 1 < : meas x 2 R3 : HðxÞjv n ðxÞjl P n n

ð3:4Þ

Passing to a subsequence if necessary, we may assume v n ! v 0 in E for some kv 0 k ¼ 1. Consequently, there exists a constant d0 > 0 such that

measfx 2 R3 : HðxÞjv 0 ðxÞjl P d0 g P d0 :

v0 2 F

since dimF < 1. It is easy to see that

ð3:5Þ

Otherwise, we have

  1 ¼ 0; meas x 2 R3 : HðxÞjv 0 ðxÞjl P n

8n 2 N;

which implies that

06

Z R3

HðxÞjv 0 jlþ2 dx 6

kv 0 k2L2 C 2 6 kv 0 k2 ! 0 as n ! 1: n n

Hence v 0 ¼ 0, which contradicts kv 0 k ¼ 1. Therefore, (3.5) holds. In view of the equivalence of any two norms on the finite dimensional space F, we have

Z R3

i.e.

jv n  v 0 j2 dx ! 0 as n ! 1;

vn ! v0 Z R3

ð3:6Þ

in L2 ðR3 Þ. By the Hölder inequality and (3.6), we get l

HðxÞjv n  v 0 j dx 6 kHk

Z 2

L2m

R3

l2 ðjv n  v 0 jÞ dx ! 0 as n ! 1: 2

ð3:7Þ

821

L. Xu, H. Chen / Applied Mathematics and Computation 243 (2014) 817–824

For each n 2 N, let



Xn ¼ x 2 R3 : HðxÞjv n ðxÞjl <

 1 ; n



XCn ¼ R3 n Xn ¼ x 2 R3 : HðxÞjv n ðxÞjl P

 1 ; n

X0 ¼ fx 2 R3 : HðxÞjv 0 ðxÞjl P d0 g: Then for n large enough, by (3.4) and (3.5), we have

measðXn

\

X0 Þ P measðX0 Þ  measðXCn

\

1 d0 P : n 2

X0 Þ P d0 

Consequently, for n large enough, there holds

Z R3

HðxÞjv n  v 0 jl dx P

Z Xn

T

1 HðxÞjv n  v 0 jl dx P l 2 X0

Z Xn

T

HðxÞjv 0 jl dx  X0

which is in contradiction to (3.7). Therefore, (3.3) holds. For the

Xu ¼ fx 2 R3 : HðxÞjuðxÞjl P kukl g;

Z Xn

T

1 d2 d0 d2 HðxÞjv n jl dx P l  0  P l0þ2 ; 2 2 n 2 X0

 given in (3.3), let

8u 2 F n f0g:

ð3:8Þ

Then by (3.3)

measðXu Þ P ;

8u 2 F n f0g:

ð3:9Þ

Combining (H4), (3.8) and (3.9), for any u 2 F n f0g, we have

BðuÞ ¼

Z

HðxÞFðx; uÞdx P

R3

Z R3

c1 HðxÞjujl dx P

Z

c1 kukl dx P 2 c1 kukl ;

Xu

which implies BðuÞ ! 1 as kuk ! 1 on any finite dimensional subspace of E. The proof is complete. h Lemma 3.2. Suppose that (H1)–(H3) are satisfied, then there exists a sequence qk ! 0þ as k ! 1 such that ak ðkÞ :¼ inf u2Zk ;kuk¼qk wk ðuÞ P 0

and

dk ðkÞ :¼ inf u2Zk ;kuk6qk wk ðuÞ ! 0 as k ! 1 uniformly for k 2 ½1; 2,

where

1

Z k ¼ aj¼kþ1 X j ¼

spanfekþ1 ; . . .g for all k 2 N. Proof. Set

l2 ðkÞ ¼ supu2Zk ;kuk¼1 kukL2 ; 8k 2 N:

ð3:10Þ

Since E is compactly embedded into L2 ðR3 Þ, there holds (see [21, Lemma 3.8].)

l2 ðkÞ ! 0 as k ! 1:

ð3:11Þ

By (H3), (3.2), (3.10), (3.11), Hölders inequality and the fact that

wk ðuÞ P

1 kuk2  k 2

Z R

HðxÞFðx; uÞdx P 3

R

KðxÞ/u u2 dx P 0, we have R3

1 1 2 m kuk2  2Cðkuk2L2 þ kukmL2 Þ P kuk2  2Cl2 ðkÞkuk2  2Cl2 ðkÞkukm : 2 2

ð3:12Þ

By (3.11), there exists a positive integer k1 such that 2

2Cl2 ðkÞ 6

1 ; 8

8k P k 1 :

ð3:13Þ

Then, by (3.12), we have

wk ðuÞ P

  1 1 3 m m kuk2  2Cl2 ðkÞkukm ¼ kuk2  2Cl2 ðkÞkukm :  2 8 8

ð3:14Þ

For each k P k1 , choose 1

qk :¼ ð16Clm2 ðkÞÞ2m :

ð3:15Þ

qk ! 0þ as k ! 1

ð3:16Þ

Then

since 1 < m < 2. By (3.14) and (3.15), direct computation shows

ak ðkÞ :¼ inf u2Zk ;kuk¼qk wk ðuÞ P

q2k 4

> 0;

8k P k1 :

822

L. Xu, H. Chen / Applied Mathematics and Computation 243 (2014) 817–824

Besides, by (3.12), for any u 2 Z k with kuk 6 qk and all k 2 ½1; 2, we have m

2

wk ðuÞ P 2Cl2 ðkÞq2k  2Cl2 ðkÞqmk : Therefore, m

2

2Cl2 ðkÞq2k  2Cl2 ðkÞqmk 6 inf u2Zk ;kuk6qk wk ðuÞ 6 0 since 1 < m < 2. For qk ! 0þ ; l2 ðkÞ ! 0 as k ! 1, we have

dk ðkÞ :¼ inf u2Zk ;kuk6qk wk ðuÞ ! 0 as k ! 1 uniformly for k 2 ½1; 2: The proof is complete. h Lemma 3.3. Under the assumptions of (H1)–(H3), then for the positive integer k1 and the sequence fqk g obtained in Lemma 3.2, there exists 0 < rk < qk for each k P k1 such that

bk ðkÞ :¼ maxu2Y k ;kuk¼rk wk ðuÞ < 0 for k 2 ½1; 2;

8k P k 1 ;

k

where Y k ¼ aj¼1 X j ¼ spanfe1 ; . . . ; ek g for all k 2 N. Proof. Note that Y k is finite dimensional space of E for each k 2 N. Then by (3.3), there exists a constant k uÞ

measðX k u

P ek ;

ek such that

8u 2 Y k n f0g; l

3

ð3:17Þ l

where X :¼ fx 2 R : HðxÞjuðxÞj P ek kuk g. Combining (3.2), (H4), (2.5) and (3.17), for any k 2 N and k 2 ½1; 2, we have

wk ðuÞ 6 6

1 C kuk2 þ kuk4  c1 2 4

Z

HðxÞjujl dx 6 R3

1 C kuk2 þ kuk4  c1 2 4

Z

HðxÞjujl dx

Xku

1 C 1 C kuk2 þ kuk4  c1 ek kukl  measðXku Þ 6 kuk2 þ kuk4  c1 e2k kukl : 2 4 2 4

ð3:18Þ

For kuk ¼ r k < qk small enough, we get

bk ðkÞ :¼ maxu2Y k ;kuk¼rk wk ðuÞ < 0 for k 2 ½1; 2;

8k P k 1 :

since 1 < l < 2. The proof is completed. Next we give the proof of Theorem 1.1. h Proof. Obviously, condition (F1) in Theorem 2.1 holds. Lemmas 3.1–3.3 show that the conditions (F2), (F3) in Theorem 2.1 hold. By Theorem 2.1, there exist kn ! 1; uðkn Þ 2 Y n such that

w0kn jY n ðuðkn ÞÞ ¼ 0; wkn ðuðkn ÞÞ ! ck 2 ½dk ð2Þ; bk ð1Þ as n ! 1:

ð3:19Þ

We claim that the sequence fuðkn Þgn2N obtained in (3.19) possesses a strong convergent subsequence in E. For the sake of notational simplicity, in what follows we always set un ¼ uðkn Þ for all n 2 N. Indeed, combining (H2), (H3), (2.1), (3.2), (3.19) and the Hölder inequality, we have

kun k2 6 2wkn ðun Þ þ 2kn

Z R3

HðxÞFðx; un Þdx 6 C 0 þ 4C

Z R3

ðHðxÞjun j2 þ HðxÞjun jm Þdx 6 C 0 þ 4Cðkun k2 þ kun km Þ

ð3:20Þ

for some C 0 > 0. (3.20) implies that fun gn2N is bounded in E since 1 < m < 2. Consequently, without loss of generality, we may assume

un * u0 as n ! 1

ð3:21Þ q

3

q

3

for some u0 2 E. Since the embedding E,!L ðR Þ is compact for any q 2 ½2; 2 Þ, then un ! u0 in L ðR Þ for any q 2 ½2; 2 Þ . By (3.2), we easily get

kun  u0 k2 ¼ ðw0kn ðun Þ  w01 ðu0 Þ; un  u0 Þ þ



Z R3

HðxÞðkn f ðx; un Þ  f ðx;u0 ÞÞðun  u0 Þdx 

Z R3

KðxÞð/un un  /u0 u0 Þðun  u0 Þdx: ð3:22Þ

Clearly

ðw0kn ðun Þ  w01 ðu0 Þ; un  u0 Þ ! 0: By (H3), Hölders inequality and Minkowski inequality, one has

ð3:23Þ

L. Xu, H. Chen / Applied Mathematics and Computation 243 (2014) 817–824

Z R3

HðxÞðkn f ðx; un Þ  f ðx; uÞÞðun  u0 Þdx  bkun  ukL2  bkun  ukL2  bkun  ukL2

Z R3

Z R3

Z R3

823

12 2 H2 ðxÞð2jun j þ ju0 j þ 2jun jm1 þ ju0 jm1 Þ dx 12 2 H2 ðxÞð2jun j þ ju0 jÞ2 þ H2 ðxÞð2jun jm1 þ ju0 jm1 Þ dx 12 ½H2 ðxÞð4jun j2 þ ju0 j2 Þ þ H2 ðxÞð4jun j2m2 þ ju0 j2m2 Þdx

3 2 12 R 12 R 2 2 2 2 þ þ 3 4H ðxÞjun j dx 3 H ðxÞju0 j dx R 7 6 R 7  bkun  ukL2 6 4 R 12 R 1=2 5 2m2 2m2 2 2 4H ðxÞjun j dx þ R3 H ðxÞju0 j dx R3 h i  Ckun  ukL2 4kun kL2 þ ku0 kL2 þ 4kun kLm21 þ ku0 kLm21 : Since un ! u0 in Lq ðR3 Þ for any q 2 ½2; 2 Þ, we obtain

Z R3

HðxÞðkn f ðx; un Þ  f ðx; u0 ÞÞðun  u0 Þdx ! 0 as k ! 1:

ð3:24Þ

By Hölder inequality, Sobolev inequality and (2.4), one gets

Z R3

KðxÞ/un un ðun  u0 Þdx 6 Ck/un kL6 kun kL3 kun  u0 kL2 6 Ck/un kD1;2 kun kL3 kun  u0 kL2 6 Ckun kL12=5 kun kL3 kun  u0 kL2 :

ð3:25Þ

Again by un ! u0 in Lq ðR3 Þ for any q 2 ½2; 2 Þ, we obtain

Z R3

KðxÞ/un un ðun  u0 Þdx ! 0 as n ! 1:

ð3:26Þ

Similarly we can also get

Z R3

KðxÞ/u0 u0 ðun  u0 Þdx ! 0 as n ! 1:

ð3:27Þ

By (3.22)–(3.27), we get kun  u0 k ! 0. Therefore, the claim above is true. Now from the last assertion of Theorem 2.1, we know that I ¼ w1 has infinitely many nontrivial critical points. Therefore, system (1.1) possesses infinitely many small negative-energy solutions by Lemma 2.1. The proof of Theorem 1.1 is complete. h

References [1] V. Benci, D. Fortunato, An eigenvalue problem for the Schrödinger–Maxwell equations, Topol. Methods Nonlinear Anal. 11 (1998) 283–293. [2] T. D’Aprile, D. Mugnai, Solitary waves for nonlinear Klein–Gordon–Maxwell and Schrödinger–Maxwell equations, Proc. R. Soc. Edinburgh Sect. A134 (2004) 893–906. [3] A. Ambrosetti, On Schrödinger–Poisson systems, Milan J. Math. 76 (2008) 257–274. [4] A. Ambrosetti, D. Ruiz, Multiple bound states for the Schrödinger–Poisson problem, Commun. Contemp. Math. 10 (2008) 391–404. [5] D. Ruiz, Semiclassical states for coupled Schrödinger–Maxwell equations concentration around a sphere, Math. Models Methods Appl. Sci. 15 (2005) 141–161. [6] Juntao Sun, Infinitely many solutions for a class of sublinear Schrödinger–Maxwell equations, J. Math. Anal. Appl. 390 (2012) 514–522. [7] Giovanna Cerami, Giusi Vaira, Positive solutions for some non-autonomous Schrödinger–Poisson systems, J. Differ. Eqs. 248 (2010) 521–543. [8] A. Azzollini, A. Pomponio, Ground state solutions for the nonlinear Schrödinger–Maxwell equations, J. Math. Anal. Appl. 345 (2008) 90–108. [9] Giuseppe Maria Coclite, A multiplicity result for the nonlinear Schrödinger–Maxwell equations, Commun. Appl. Anal. 7 (2003) 417–423. [10] Hiroaki Kikuchi, On the existence of a solution for elliptic system related to the Maxwell–Schrödinger equations, Nonlinear Anal. 67 (2007) 1445–1456. [11] S. Chen, C. Tang, High energy solutions for the Schrödinger–Maxwell equations, Nonlinear Anal. 71 (2009) 4927–4934. [12] Q. Li, H. Su, Z. Wei, Existence of infinitely many large solutions for the nonlinear Schrödinger–Maxwell equations, Nonlinear Anal. 72 (2010) 4264– 4270. [13] Juntao Sun, Haibo Chen, Liu Yang, Positive solutions of asymptotically linear Schrödinger–Poisson systems with a radial potential vanishing at infinity, Nonlinear Anal. 74 (2011) 413–423. [14] Minbo Yan, Zifei Shen, Yanheng Ding, Multiple semiclassical solutions for the nonlinear Maxwell–Schrödinger systems, Nonlinear Anal. 71 (2009) 730– 739. [15] Qing dong Li, Han Su, Zhongli Wei, Existence of infinitely many large solutions for the nonlinear Schrödinger–Maxwell equations, Nonlinear Anal. 72 (2010) 4264–4270. [16] A. Azzollini, P. d’Aveniab, A. Pomponio, On the Schrödinger–Maxwell equations under the effect of a general nonlinear term, Ann. I.H.Poincarr´AN27(2010) 779–791. [17] W. Zou, Variant fountain theorems and their applications, Manuscripta Math. 104 (2001) 343–358. [18] Q. Li, H. Su, Z. Wei, Existence of infinitely many large solutions for the nonlinear Schrödinger–Maxwell equations, Nonlinear Anal. 72 (2010) 4264– 4270. [19] T. Bartsch, Z. Wang, Existence and multiple results for some superlinear elliptic problems on RN , Commun. Partial Differ. Equ. 20 (1995) 1725–1741.

824 [20] [21] [22] [23] [24]

L. Xu, H. Chen / Applied Mathematics and Computation 243 (2014) 817–824 T. Bartsch, A. Pankov, Z.Q. Wang, Nonlinear Schrödinger equations with steep po tential well, Commun. Contemp. Math. 3 (2001) 549–569. M. Willem, Minimax Theorems, Birkhäuser, Boston, 1996. V. Benci, D. Fortunato, A. Masiello, L. Pisani, Solitons and the electromagnetic field, Math. Z. 232 (1999) 73–102. L.C. Evans, Partial Differential Equations, American Mathematical Society, Providence, RI, 1998. K. Benmlih, Stationary solutions for a Schrödinger–Poisson system in R3 , in: Proceedings of the 2002 Fez Conference on Partial Differential Equations, Electron. J. Differ. Equ. Conf., vol. 9, 2002, pp. 65–76.