Existence of Solutions for a Schrödinger–Poisson System with Critical Nonlocal Term and General Nonlinearity

Abstract

We study the existence and multiplicity of nontrivial solutions for a Schrödinger–Poisson system involving critical nonlocal term and general nonlinearity. Based on the variational method and analysis technique, we obtain the existence of two nontrivial solutions for this system.

1. Introduction and Main Result

The Schrödinger–Poisson system is usually used to describe solitary waves for the nonlinear stationary Schrödinger equations interacting with an electromagnetic field. Since the introduction of the Schrödinger–Poisson system by Benci and Fortunato [1], it has been extensively studied. For more detailed information, we refer the interested readers to [2, 3] and the references therein.

In recent years, some researchers extensively studied the Schrödinger–Poisson system with critical growth in an unbounded domain and obtained interesting results under various suitable assumptions (see, e.g., [4–10]).

But there are currently only a few results for the following Schrödinger–Poisson systems with critical nonlocal terms in a bounded domain [11–13]: where ; are real numbers; and is a continuous function satisfying some suitable assumptions. In [11], assuming that and , the author proved that system (1) has a positive ground state solution for any and , where is a real number and is the first eigenvalue of . Later, when , where is a real number, the authors in [12] studied system (1); they proved existence and nonexistence results of positive solutions when and existence of solutions in both the resonance and the nonresonance case for higher dimensions. For the case is a real number, , and , in [13]; when , authors proved that system (1) has at least two positive solutions if for some small enough, and when , system (1) has at least one positive solution for any .

On the basis of the above literature, this paper continues to study system (1), and intends to deal with the following Schrödinger–Poisson system with critical nonlocal term and general nonlinearity that without (AR) condition: where is an open bounded domain with smooth boundary ,

is the critical Sobolev exponent, and .

Throughout this paper, we make the following assumptions:

(g₁) , and there exist constants with which is small enough and such that .

(g₂) There exists a constant big enough such that for any and large enough.

(g₃) There are constants and such that

The main difficulties in the present paper are to estimate the critical value and prove the boundedness of (PS) sequence due to the lack of compactness. In order to overcome the above difficulties, by analytic techniques, we shall give the estimate of critical value of associated functional so that system (2) has at least two nontrivial solutions.

Throughout this paper, we use the following notations: (i)The space has the inner product and the norm , and the norm in is denoted by (ii) (respectively, ) denotes the closed ball (respectively, the sphere) of center zero and radius r, i.e., (iii),... denote various positive constants, which may vary from line to line(iv)Define the best constant which is attained by the functions for all , where .

Theorem 1. Assume that satisfies (g₁), (g₂), and (g₃). Then, system (2) possesses at least two distinct nontrivial function pair solutions.

Remark 2. Relative to [11, 12], the nonlinearity is of a pure power form in [11, 12], and in the present paper, it is a general nonlinearity. Hence, we make a substantial improvement on the works of the former.

2. Proof of Main Result

Before proving our Theorem 1, we need the following lemmas.

Lemma 3 ([12, 13]). For every fixed , there exists a unique that solves the second equation of (2), and (i)(ii)For all (iii)(iv)

Hence, according to the standard arguments as those in [1], system (2) can be converted into the following boundary value problem:

In order to study the existence of nontrivial solutions to problem (3), we shall firstly consider the existence of nontrivial solutions of the following problem: where

The energy functional corresponding to (4) is where

is well defined with and

The critical points of the functional are just weak solutions of problem (4). Let define a cutoff function such that where for

Put ; hence, .

Lemma 4 ([14, 15]). satisfies the following estimates:

Lemma 5. Assume (g₁) and (g₃) hold; let be a sequence such that , where. Then, there exists such that , up to a subsequence. and is a nontrivial solution of problem (4).

Proof. First, we prove that is bounded in . To prove the boundedness of , arguing by contradiction, suppose that . Set . Then, and for . By (g₃), we have where , which implies

Passing to a subsequence, we may assume that in , then in , and a.e. in . Hence, it follows from (13) that , and where is chosen such that and is sufficiently big constant, which is a contradiction. Thus, is bounded in and there exists such that , up to a subsequence. Furthermore, by the weak continuity of . If in , since the term is subcritical, then implies

By Lemma 3-(iii), one has

It follows from (15) and (16) that

If , it contradicts . Therefore,

By (15) and (18), we get which contradicts . Thus, and it is a nontrivial solution of problem (4).

Lemma 6. Assume that satisfies (g₁) and (g₂). Then, for small enough, .

Proof. For , we consider the functions where the above inequality comes from Lemma 3-(iv).

Notice that , and as is sufficiently small. Therefore, is attained for some . Since we have where the nonnegativity of comes from (g₂), (g₃), and the definition of . Hence,

It follows from (g₁) that

Hence, we have

By (10), (11), and (25), when is sufficiently small, we have with .

On the other hand, the function attains its maximum at and it is increasing in the interval . By (10), (25), and for , we deduce that

From (11) and the fact that is sufficiently large, by choosing sufficiently small , we can obtain

Proof of Theorem 1. It follows from (g₁) that for all and . By the Sobolev inequality, (28) and (29), for small enough, one has

So, when is sufficiently small, there is such that for . Moreover, by the nonnegativity of , for , it holds that

Hence, we can choose such that and . Using the Mountain Pass Lemma, there is a sequence satisfying where

According to Lemma 5 and Lemma 6, we can get a sequence , and such that . Thus, is a solution of problem (4). And then, , where . Hence, , that is, . We conclude from the strong maximum principle that is a positive solution of problem (3).

Next, we give the proof of two nontrivial solutions to system (2).

From the above discussion, problem (3) has a positive solution . Put for. Note that if is a solution of (3), then, is a solution of (3) replacing with . Hence, the equation has at least a positive solution v. Let ; then, is a solution of (3).

Obviously . So, problem (3) has at least two distinct nontrivial solutions and ; therefore, similar to [4, 5], system (2) has at least two distinct nontrivial function pair solutions and .

Data Availability

The findings in this research do not make use of data.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

This paper is supported by the National Natural Science Foundation of China (Nos. 11661021 and 11861021); the Science and Technology Foundation of Guizhou Province (No. QKH[2017]1084); the Young Science and Technology Scholars of Guizhou Provincial Department of Education (No. KY[2016]164); and the Key Laboratory of Advanced Manufacturing Technology, Ministry of Education, Guizhou University (No. KY[2018]479).