Multiplicity Results for a Class of Kirchhoff-Schrödinger-Poisson System Involving Sign-Changing Weight Functions

Abstract

This paper deals with the Kirchhoff-Schrödinger-Poisson system involving sign-changing weight functions. We prove the existence and multiplicity of solutions to the system. Our main results are based on the method of Nehari manifold.

1. Introduction

In this paper, we consider the following Kirchhoff-Schrödinger-Poisson system with sign-changing weight functions:where , , , , , and .

We may assume that the weight functions and satisfy the following conditions:

(H1) and where , and in addition a.e. in in case .

(H2) , and , where .

When , system (1) is the following Kirchhoff-type equation: The Kirchhoff-type equation was first put forward by Kirchhoff [1] as an extension of the classical D’Alembert wave equation for free vibrations of elastic string. Since the Kirchhoff-type problems arise in various models of physical and biological systems, many researchers have studied these problems in recent years; see [2–8] and references therein.

Recently, in [9], using the symmetric mountain pass theorem, Zhao, Zhu, and Li investigated the existence of infinitely many solutions to system (1) with the nonlinearity being replaced by , in which has sublinear growth in .

When , , and the nonlinearity is replaced by , system (1) reduces to the following Schrodinger-Poisson system: which has been first introduced in [10] as a physical model describing solitary waves interacting with its own electrostatic field in quantum mechanics. For more physical background of the system (3), we refer the readers to [11–13] and the references therein. In recent years, a great deal of work has been done in the study of (3) with and via variational methods and critical point theory under various hypotheses on the potential and the nonlinear term ; see [14–20] and the references cited there.

Motivated and inspired by [9, 21, 22] and the aforementioned works, in this paper, we investigate the existence and multiplicity of solutions to system (1) involving sign-changing weight functions. The main results we get are based on Nehari manifold. This article is organized as follows. In Section 2, we give some notations and preliminaries. Section 3 is devoted to the proof of our theorems.

2. Preliminaries

Let be the Sobolev space equipped with the norm . Let with the norm .

Define For , , the following hypotheses hold:

(H3) , . For every , the set has finite measure.

(H4) , and .

It is known that from (H3) is a Hilbert space with the inner product and endowed with the norm

Lemma 1 (see [9]). By conditions (H3) and (H4), is continuously embedded into Thus, the embedding is continuous for , and then there exists such that

Lemma 2 (see [9]). For any , there exists a unique which solves the following equation: Moreover, for any , can be expressed in the following form:

Lemma 3 (see [9]). For any , we have (i) and .(ii).(iii)If in , then in and in .

Now, we define the energy functional associated with problem (1) by By [9], we know that is well defined and with Obviously, if is a critical point of , then the pair is a solution of system (1).

The best Sobolev constant is defined as follows:

The Nehari manifold for is defined as

The Nehari manifold is closed linked to the behavior of functions of the form for named fibering maps [23]. If , we have and Clearly, which implies that for and , if and only if ; i.e., positive critical points of correspond to points on the Nehari manifold. In particular, if and only if . Therefore, define

For each , we have

If , then , and it follows from (17) and (18) that and

Lemma 4. If (H1)-(H4) hold, then the energy functional is coercive and bounded below on .

Proof. For , we have by Holder and Sobolev inequalities that where . Thus is coercive and bounded below on .

Lemma 5. Assume that (H1)-(H4) hold. There exists such that, for any , we have .

Proof. If not, that is for each , then by (19) and the Holder and Sobolev inequalities we have for that which implies that and so On the other hand, we obtain by (20) and the Holder and Sobolev inequalities that which implies that If is sufficiently small, then (24) contradicts with (26). Thus, we conclude that there exists such that for .

Let

From Lemma 5, for , we have and define

Lemma 6.  (i)If , then .(ii)If , then .

The proof is an immediate result from (19) and (20).

Define the function as follows: Clearly, if and only if . Furthermore, which implies that for . That is, (or ) if and only if (or ).

Set

Lemma 7. Assume that (H1)-(H4) hold. Let . Then for each and , we have the following.
(1) If , then there exists a unique such that and (2) If , then there exists a unique such that , and

Proof. From (29) and (30), we have that , as , and . Thus there exists a unique such that is increasing on and decreasing on and . Moreover, is the root of which implies that From (35), we obtain Thus, we have by (35), (36), and the Holder and Sobolev inequalities that Case 1 []. Then has unique solution and . On the other hand, we have and Therefore, or . For , we obtain for . Thus, . Furthermore, we have Let Similar to the argument in the function , we get that achieves its maximum at . Thus, we have Case 2 []. By (37), Then there exist and such that , Moreover, we have and . Thus, there are two multiplies of lying in , that is, and , and for each and for each . Hence, and

Let

(H5) , and .

Remark 8. If , then , which implies that .

Lemma 9. Assume that conditions (H1)-(H5) are satisfied; then we have .

Proof. For , we obtain Hence, we have which implies that .

3. Main Results

Using the idea of Ni-Takagi [24], we obtain the following Lemma.

Lemma 10. For each , there exist and a differentiable function such that , the function , and for all , where

Proof. For , we define a function by Then and From the implicit function theorem, we know that there exist and a differentiable function such that , where is as in (50), and which is equivalent to which implies that .

By means of the similar technique used in Lemma 10, we have the following consequence.

Lemma 11. For each , there exist and a differentiable function such that , the function , and for all , where is as in (50).

Let

Lemma 12. Assume that (H1)-(H5) hold. If , then for ,
(i) there exists a minimizing sequence such that (ii) there exists a minimizing sequence such that

Proof. By the Ekeland variational principle [11] and Lemma 5, there exists a minimizing sequence such that andLet be large enough; by Lemma 9, we get which implies that This implies and by using (61), (63), and the Holder inequality, we obtain and In the following, we will prove that In fact, by using Lemma 10 with to get the functions for some , such that , fixed and we choose . Let with and let . Set ; since , we deduce from (61) that and by the Mean Value Theorem, we get Therefore, By and (69) it follows that Hence, Since andtaking limit in (71), we obtain for some constant , independent of . In the following, we will show that is uniformly bounded in . From (49), (65), and the Holder inequality, we obtain for some that We only need to prove that for some and large enough. If (76) fails, then there exists a subsequence such that Combining (77) with (64), we may find a suitable constant such that By (77) and , we have Furthermore, we obtain by (77) and (79) that which implies that Let whereIn view of (77), it is easy to know that Thus, But, by (78), (81), and , which contradicts (85). Hence, we obtain This completes the proof of (i). Similarly, we can prove (ii) by using Lemma 11.

Theorem 13. Assume that (H1)-(H5) hold. For each ( is as in Lemma 12), the functional has a minimizer in satisfying that (1);(2) is a solution of (1).

Proof. By Lemma 12 (i), there exists a minimizing sequence for on such that From Lemma 9 and the compact imbedding theorem, we know that there exist a subsequence and such that and In the following we will prove that . In fact, if not, by (90) and the Holder inequality we can get that as . Hence, and which contradicts as . Moreover, Hence, is a nonzero solution of (1) and . Next, we will prove that . By Lemma 5, we get that By using in [9], we obtain that Combining Fatou’s lemma and (96), we have Hence, . Moreover, we have . In fact, if , by Lemma 7, there are unique and such that and ; we have . Since there exists such that . By means of Lemma 4, we get which is a contradiction. Since and , we have that is a solution of (1) in virtue of Lemma 6.

Similarly, we can obtain the theorem of existence of a local minimum for on as follows.

Theorem 14. Assume that (H1)-(H5) hold. For each , the functional has a minimizer in satisfying that (1);(2) is a solution of (1).

Finally, we give the main result of this paper as follows.

Theorem 15. Suppose that conditions (H1)-(H5) hold. Then there exists such that for , (1) has at least two solutions.

Proof. From Theorems 13 and 14, we have that (1) has two solutions and such that , . Since , we get that and are different, which completes the proof.

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that there are no conflicts of interest.

Authors’ Contributions

The authors contributed equally to this paper. All authors read and approved the final manuscript.

Acknowledgments

This work is supported by Natural Science Foundation of China (11571136).