Abstract

In this paper, we study the existence of positive solutions for Schrödinger-Poisson systems with sign-changing potential and critical growth. By using the analytic techniques and variational method, the existence and multiplicity of positive solutions are obtained.

1. Introduction and Main Result

In this paper, we consider the existence and multiplicity of positive solutions of the Schrödinger-Poisson systems with critical nonlinearity of the following form: where is a smooth bounded domain in , , , and . When , , the right hand side of the system of (1) has the critical term . It is pointed out that system (1) is related to the following Schrödinger-Poisson system: where the potentials , satisfy some mild conditions. The existence, uniqueness, and multiplicity of positive solutions of systems like (2) have been extensively studied in the last few decades, such as [1–14]. Furthermore, the Schrödinger-Poisson systems involving critical growth have been attracted many researchers, e.g., [15–24]. But there is little literature for liking the system (1) that considers the critical growth on bounded domains, such as [15–17].

In [24], the authors considered a Schrödinger-Poisson system with concave-convex nonlinearities of the form where is a smooth bounded domain in , and . They obtained two positive solutions when is enough small by using the concentration-compactness principle.

Recently, in [25], the authors considered a Schrödinger-Newton system with critical nonlinearities of the form where is a smooth bounded domain in , , , and . The authors studied the existence and multiplicity of positive solutions by the variational method and analytic techniques.

Motivated by results found in the above mentioned papers, in the present paper, we intend to study the existence of multiple positive solutions for system (1). Since system (1) contains a critical term and a nonlocal term, the critical value level becomes very difficult to estimate. The main objective of this paper is to look for new estimates and establish two positive solutions by analytic techniques.

The main result is as follows:

Theorem 1. Assume that and is a possibly sign-changing function. Then, there exists such that for any , system (1) has at least two different positive solutions. Additionally, system (1) admits a positive ground state solution.
Throughout this paper, we use the following notations: (i)The space is equipped with the norm . The norm in is denoted by (ii) denote various positive constants, which may vary from line to line(iii)We denote by (respectively, ) the closed ball (respectively, the sphere) of center zero and radius , i.e., , (iv)Let be the best Sobolev embedding constant for the embedding , namely,

2. Existence of a First Positive Solution of System (1)

For any given , by the Lax-Milgram theorem, the Dirichlet boundary problem in has a unique solution . Substitute in the first equation of system (1), system (1) transformed into the following variable equations:

The formal energy functional corresponding to the system (6) is the following:

If a function satisfies where , we can prove that is a solution of system (1) if and only if , and is called a weak solution of (6).

To prove our main theorem 1, some preliminary results are needed, let us first collect some properties.

Lemma 2 (see [17]). For each , there exists a unique solution of Moreover, (1) for (2)For each , (3)(4)Suppose that in , then

Lemma 3 (see [18]). Let , , and , then there exists such that .

Lemma 4. There exist , such that every , we have

Proof. According to the Sobolev and Hölder inequalities and Lemma 2, we have Setting for , there exists a positive constant such that . Letting . Consequently, it follows that for any .
Additionally, by Lemma 3, there exists such that For small enough, we have . The lemma is proved.

We recall that for , a sequence is called a sequence for if and strongly in the dual space of as . That is to say, satisfies the condition in if every sequence in for contains a convergent subsequence.

Lemma 5. If is a sequence for with where . Then, there exists such that in and .

Proof. Let be such that We claim that is bounded in . Indeed, by (15), we have that which implies that is bounded in . Thereby, there exists a subsequence, still denoted by , and a function , such that as . Set , then . Otherwise, there exists a subsequence (still denoted by ) such that By (15), letting , for every , it follows In particular, taking the test function in (19), we have By Brézis–Lieb (see [26]) and Lemma 2, one has Therefore, according to as , we have It follows from (20) and (22) implies that Note that , we deduce that On the one hand, by using the Young inequality and (20), there holds where is a constant.
On the other hand, note that , by (23), one has From the above analysis, this is a contradiction. Therefore, , it implies that in , and So as . The lemma is proved.

Theorem 6. Assume and . Then, system (1) has a positive solution which satisfies .

Proof. From Lemma 4 and applying Ekeland’s variational principle in , there exists a minimizing sequence such that Therefore, Since is bounded in , there exist and a subsequence still denoted by such that in as .
Note that and Lemma 5, we can obtain in and and . Since , the embedding theorem means . Note that , by the regularity of weak solutions, one has . Thereby, according to the classical bootstrap argument, we have for . By the embedding theorem again, it holds that for . Since , we have for some . Thereby, by the strong maximum principle, we obtain in , then is a positive solution of system (1) with (by (11)). The theorem is proved.
For , we have Now, if we consider the following problem: Then, we find the weak solutions of problem (32) correspond to the critical points of the functional , where Obviously, one has