On the Existence of Solutions for a Class of Schrödinger–Kirchhoff-Type Equations with Sign-Changing Potential

Abstract

In this paper, we consider the following Kirchhoff problem where are constants, is a positive parameter, and . Under suitable assumptions on , the existence of nontrivial solution is obtained via variational methods. The potential is allowed to be sign-changing.

1. Introduction and Main Results

In this paper, we consider the following Kirchhoff type problem:where are constants, is a positive parameter, , and the potential satisfies the following conditions:   and is bounded below  there exists a constant such that the set is nonempty and , where denote the Lebesgue measure in

This kind of assumptions was first introduced by Bartsch and Qiang Wang [1] in the study of the nonlinear Schrödinger equations and has attracted the attention of several researchers.

In recent years, the Kirchhoff problem on a bounded domain has been studied by many authors (see, for example, [2–8]). More recently, many researchers focused on the Kirchhoff problem defined on the whole space , i.e., the following problem:where is a potential function and . In [9], Wu studied (3) by using a symmetric Mountain Pass Theorem under the following assumptions about potential

, , where is a constant. Moreover, for any , , where denotes the Lebesgue measure in .

Under this condition, by Lemma 3.4 in [10], the embedding is compact for any . Hence, the corresponding results in [9] have been obtained by using the variational techniques in a standard way. In [11–13], the authors considered Kirchhoff type problem (3) with a steep potential well. Precisely, the potential function satisfies the following conditions besides :   and on   is a nonempty open set with locally Lipschitz boundary and

By using this conditions, Sun and Wu [11] considered (3) in the case where the nonlinearity is asymptotically -linear with respect to at infinity. Du et al. [12] studied (3) when behaves like with and proved the existence and asymptotic behavior of ground state solutions. Zhang and Du [13] investigated the existence and asymptotic behavior of positive solutions for (3) by combining the truncation technique and the parameter-dependent compactness lemma for small and large in the case where behave like with . For more results about Kirchhoff type problems, we refer the reader to [14–18] and the references therein.

Under the assumption of , the potential may change sign. The purpose of this paper is to consider the multiplicity of solutions for (1) in this case. To our best knowledge, there is no existence result of solutions for (1) with sign-changing potentials. Our main result as follows.

Theorem 1. Suppose that and and hold. Then, system (1) possesses infinitely many distinct pairs of nontrivial solutions whenever is sufficiently large.

2. Preliminaries

As a matter of convenience, without loss of generality, we may assume that and . Consequently, we are dealing with the Kirchhoff type problem as

Letbe the usual Sobolev space with the standard inner product and norm as follows:

In our problem, we work in the space defined bywith the inner product and the norm as follows:where and . It follows from the conditions and the Hölder and Sobolev inequalities thatwhich implies that the embedding is continuous. Here, is the best constant for the embedding of in . Combine with the continuity of the following embedding:

There is a constant such that

As a consequence, the functional given byis well defined, and it is of class with derivativefor all . As in [19], letand denote the orthogonal complement of in by . Consider the eigenvalue problem

In view of and , the quadratic form is weakly continuous. We have the following proposition.

Proposition 1 (see Lemma 2.1 in [19]). Suppose and and hold. Then, for each fixed ,  as   is a non-increasing continuous function of  where is sequence of positive eigenvalues of problem satisfying as and the corresponding eigenfunctions .LetThen,

Moreover, dim for every fixed .

To complete the proof of our theorem, we need the following results.

Theorem 2 (see Theorem 9.12 in [20]). Let be an infinite dimensional Banach space, and let be even, satisfying (PS) condition and . If , where is finite dimensional and satisfies the following:  there are constants such that   for each finite dimensional subspace , there is an such that on  then possesses an unbounded sequence of critical values.

3. Proof of Main Results

Lemma 1. Suppose that and and hold. Then, there exist such that for all with .

Proof. By Proposition 1, for each fixed , there exists a positive integer such that for and for . Thus, for any , we havefor all , where . Since , the conclusion follows by choosing sufficiently small.

Lemma 2. Suppose that and hold. Then, there is a large such that on .

Proof. Since all norms are equivalent in a finite dimensional space, there are constants and such thatwhere . Hence, for all ,Since , consequently, there is a large such that on .

Lemma 3. Let and be satisfied. Then, there exists such that, for each , satisfies the condition for all .

Proof. Let be a sequence, that is, and . If is unbounded in , up to a subsequence, we can assume thatas , after passing to a subsequence. Set , we can assume that in and a.e. .
If , since is weakly continuous, we havea contradiction. If , then the set has positive Lebesgue measure. For , one has as ; Fatou’s lemma shows that as . Thus, by (9), we obtainThis is a contradiction. This implies is bounded in . We assume that . Passing to a subsequence if necessary, we can assume that there exists and such thatThen, implies thatTaking in (25), we obtainLet . It follows from thatMoreover, Let . Then, . By Sobolev inequalities and Hölder inequality, one haswe knowLetting be so large that the term in the brackets above is positive when , we get in . Since and , it follows that in . This completes the proof.

Proof of Theorem 1. Obviously, . Furthermore, is even. The conclusion follows from Lemmas 1–3 and Theorem 1.

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

This work was supported partially by the National Natural Science Foundation of China (11961078) and Foundation of Baoshan University (BYPY202016).