Existence of Multiple Nontrivial Solutions for a Strongly Indefinite Schrödinger-Poisson System

Abstract

We consider a Schrödinger-Poisson system in with a strongly indefinite potential and a general nonlinearity. Its variational functional does not satisfy the global linking geometry. We obtain a nontrivial solution and, in case of odd nonlinearity, infinitely many solutions using the local linking and improved fountain theorems, respectively.

1. Introduction and Statement of Results

In this paper, we consider the Schrödinger-Poisson system:

For , , , , and , we assume the following. and for some . This assumption ensures that the Schrödinger operator is self-adjoint and semibounded on (see Theorem A.2.7 in [1]). denotes the spectrum of . We assume that lies in a gap of ; that is, there exist such that satisfy for all ,  and there exists such that and there exists such that for all ,

as .Let  We assume that there exists such that for all , for all .

Remark 1. Note that This together with (7) implies that there exists such that for all ,

Our main results are as follows.

Theorem 2. Suppose that , , and are satisfied, then the problem (1) has a nontrivial solution.

Theorem 3. Suppose that , , and are satisfied, then the problem (1) has infinitely many solutions.

Problem (1) arises in quantum mechanics and is related to the study of the nonlinear Schrödinger equation for a particle in an electromagnetic field or the Hartree-Fock equation. For a more detailed physical background of the Schrödinger-Poisson system, readers can refer to [2, 3] and the references therein.

This system has attracted considerable research attention in the recent decade, and it has been studied widely by using the modern variational method and critical point theory under various assumptions. However, many mathematical studies have been devoted to the case . In this case, there are many results on the existence, nonexistence, or multiplicity of solutions for (1). One can refer to [2–20].

There are very few studies devoted to (1) under the assumption that has a nontrivial negative eigenspace compared to the case . In a recent paper [21], Chen and Liu studied the problem under the assumption on that() is bounded from below and for every , where is the Lebesgue measure on . Moreover, the operator has negative eigenvalues.

They verified the existence of multiple solutions of (1) under this assumption on and under certain 4-superlinear conditions on . Our assumptions on , which are different from , allow an infinitely dimensional negative eigenspace of . This causes some difficulties. For example, it makes the verification of the compactness conditions a more delicate problem. In addition, when we search for infinitely many solutions of (1) for the case where is odd, the classical fountain theorem of Bartsch (see [22] or [23]) cannot be applied. Fortunately, this difficulty can be overcome using a recently improved fountain theorem of Batkama and Colin [24]. To the best of our knowledge, the Schrödinger-Poisson equation with a strongly indefinite linear part has never been studied. Besides the difficulties caused by the strongly indefinite linear part, the functional related to (1) (see Section 2) involves a nonlocal term and it makes the functional not satisfy the global linking structure. To overcome this difficulty and obtain a nontrivial solution of (1), we use the local linking method (see [25]).

Throughout this paper, we denote the strong and the weak convergence by and , respectively. denotes the standard Lebesgue space with norm . For , denotes the standard Sobolev space with norm . For a Banach space , we denote the dual space of by , and the norm of is denoted by.

2. Proof of Theorem 2

Assume that holds and let be the self-adjoint operator acting on with domain . By virtue of , we have the orthogonal decomposition such that is negative (resp. positive) in (resp. in ). Let be equipped with the inner product and norm , where denotes the inner product of . From , with equivalent norms. Therefore, continuously embeds in for all . In addition, we have the decomposition where is orthogonal with respect to both and . Therefore, for every , there is a unique decomposition with and

For , it is well known (see, e.g., Theorem of [26]) that the Poisson equation has a unique solution and .

Let Under the assumptions and ; is a functional in . The derivative of is given by It is easy to see that if is a critical point of , then is a solution of (1).

Our functional does not satisfy the geometric assumptions of the generalized linking theorem (see, e.g., [23, Chapter 6]) because of the term To overcome this difficulty, we apply the local linking theorem to find critical points of .

Recall that by definition (see [25]), a functional defined in has a local linking at with respect to the direct sum decomposition , if there is such that Let be the total orthonormal sequences in . We consider two sequences of finite dimensional subspaces where . It is easy to see that For , let and denote the restriction of on .

Definition 4. We say that satisfies condition if any sequence such that contains a subsequence that converges to a critical point of .

From [27, Theorem 2.2], we have the following.

Theorem 5. Suppose that has a local linking at , satisfies condition, maps bounded sets into bounded sets, and for every , Then, has a nontrivial critical point.

Lemma 6. The functional has a local linking in with respect to the direct sum decomposition .

Proof. From the Hardy-littlewood-Sobolev inequality (see, e.g., [28]), we infer that there exists such that for every , From , we deduce that is bounded in , . Therefore, by the Sobolev inequality, we have It follows that for some .
From and , we deduce that for any , there exists such that This together with the fact that is a bounded function in (see ) implies that there exists such that Combining (28), (30), and the definition of (see (18)), we get that for any , and for any , Choose . Then, from the above two inequalities, we deduce that we can choose small such that satisfies (21). Therefore, has a local linking in with respect to the direct sum decomposition .

Lemma 7. Under the assumptions ,, and , the functional satisfies the condition.

Proof . Let be a sequence; that is, and .
First, we prove that is bounded in . From (7), we have From (26), we have By the Hölder inequality and (4), we have Combining (34) and (35), we get that This together with (33) yields that is bounded.
Second, we prove that is bounded. We have From (5) and the Hölder inequality, we get that for any Since is bounded, From (see ) and the Sobolev inequality, we get that there exists such that From the Hölder and Sobolev inequalities and boundedness of , we have where is a positive constant that depends only on . Combining (38)–(41), we infer that there exists such that
From the Hardy-Littlewood-Sobolev and Hölder inequalities, we get that From (43), (35), and the boundedness of , we get that there exists a constant such that Similarly, we have Combining (37), (42), (44), and (45), we infer that is bounded.
Finally, we prove that has a convergent subsequence. Up to a subsequence, we assume that in . Then, is a critical point of . From , and from and , Therefore,
By and , we get that Similarly, By (48)–(50), , and we get that in . The same argument implies that in . Therefore, in .

Remark 8. From the proof of this theorem, we infer that also satisfies the Cerami condition; that is, if satisfies then contains a convergent subsequence.

Lemma 9. The functional satisfies (25).

Proof. If the functional does not satisfy (25), then there exist , and such that and This together with (36) and (9) yields Since , there exists such that Then, by (54), We use to denote the closure of under the norm . Since there exists a continuous projection , there exists such that for every , Since and is finite dimensional, from (56) and (57), we get that This contradiction implies that satisfies (25).

Proof of Theorem 2. The desired result of Theorem 2 can be derived by combining Theorem 5 and Lemmas 6, 7, and 9.

3. Proof of Theorem 3

Recall that are the total orthonormal sequences in . For , let

Let , be the orthogonal projections. As [23], on , we define The topology generated by will be denoted by and all topological notations related to it will include this symbol.

For the proof of Theorem 3, we use the following improved fountain theorem of Batkam and Colin [24, Theorem 12], which is a generalization of the classical fountain theorem of Bartsch [22] (see also [23]).

Theorem 10. Assume that an even functional satisfies the following:(A) is -upper semicontinuous, and the gradient is weakly sequentially continuous.If there exist such that(i), as ,(ii), then there exist and sequences such that for every , and

Remark 11. In [24], the result of this fountain theorem is Since the deformation theorem is still valid under the Cerami condition (see, e.g., [27]), replacing the pseudogradient vector field in page 442 of [24] by the Cerami-type pseudogradient vector field , we see that similar to many critical point theorems, the result of the fountain theorem in [24] can be improved as (61).

Proof of Theorem 3. (1) Let us prove that is -upper semicontinuous. Assume that and . Since , are bounded so that , it follows that in . Since , from and , we obtain In addition, from the proof of (49) we infer that Using the Fatou lemma, we obtain
(2) The proof that is weakly sequentially continuous is similar to that in the proof of Lemma 6.15 of [23].
(3) Verification of (i) for : since , we infer that if in , then It follows that where From and , we deduce that for any , there exists such that For , by the Sobolev inequality and (67), Choosing and letting , we get from (70) that for , Since and , it follows that
(4) Verification of (ii) for : since , (ii) is a direct consequence of Lemma 9.
Finally, from , we obtain that is an even functional. Then, from (1)–(4) and Remark 8, we deduce that for every , there exists a critical point of such that . Therefore, (1) has infinitely many solutions .

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

The authors would like to thank the anonymous referees for their comments and suggestions on the paper. Shaowei Chen was supported by Science Foundation of Huaqiao University and Promotion Program for Young and Middle-aged Teacher in Science and Technology Research of Huaqiao University.