The Existence of the Sign-Changing Solutions for the Kirchhoff-Schrödinger-Poisson System in Bounded Domains

Abstract

In this paper, we study a class of the Kirchhoff-Schrödinger-Poisson system. By using the quantitative deformation lemma and degree theory, the existence result of the least energy sign-changing solution is obtained. Meanwhile, the energy doubling property is proved, that is, we prove that the energy of any sign-changing solution is strictly larger than twice that of the least energy. Moreover, we also get the convergence properties of as the parameters and .

1. Introduction and the Main Results

In this paper, the following Kirchhoff-Schrödinger-Poisson system is considered: where is a bounded domain with a smooth boundary , , and satisfies some basic assumptions.

For , problem (1) reduces to the following Schrödinger-Poisson system:

Alves and Souto [1] studied the above Schrödinger-Poisson system for . Under some suitable assumptions on the nonlinearity , by using the deformation lemma and Brouwer’s topological degree theory, they proved that the above system possessed a least-energy sign-changing solution, which changed sign only once.

For , the problem (1) reduces to the following problem:

The problem (3) has been studied in [2, 3]. Under different assumptions on , the authors in [2, 3] obtained the existence and some qualitative properties of the sign-changing solution by using the Non-Nehari manifold method and deformation lemma. We can find that the results in [3] improve and generalize the results in [2]. In fact, the studies about the existence of the positive solutions, sign-changing solutions for a class of elliptic equations, have been studied extensively. For more details about such problems, we refer the reader to [4–19].

To our best knowledge, the results of the sign-changing solutions for the Kirchhoff-Schrödinger-Poisson system under a weak assumption that have not been studied yet. This paper attempts to fill this gap in the literature. Motivated by the above papers, we study the problem (1). For the purpose of getting the results, we use the variational method and some mathematical skills to obtain the existence of the sign-changing solution and its corresponding properties.

In this paper, we assume satisfies the following four conditions:

,, where is a constant

there exists a such that for any and , where is the first eigenvalue for the following problem:

Throughout this paper, we will use the following notations.

Let be the usual Sobolev space equipped with the following norm:

The usual norm is denoted by . In this way, we know .

For the Poisson system, where there exists a unique , such that satisfies the above system. It is known that satisfies the following conditions [17–19]: (i)(ii) and for (iii)for and for (iv)if in , then in and if in , then

Consequently, is a solution of problem (1), that is, and are a solution of the following problem:

In this paper, is called a solution for problem (1), which implies is a solution of problem (1).

Next, we can define the energy functional corresponding to problem (1) by where . Obviously, the functional is well defined and belongs to . By a simple computation, we have that for any ,

It is clear that the critical points of are the weak solutions for the problem (1). If is called a sign-changing solution of problem (1), then and for any , where .

For and , by (9) and (10), we have

To get the main results, we restrict in the following sets:

To get the energy doubling property, we define and

To prove the convergence property, we give the following definitions. Firstly, we define the energy functional corresponding to (2) by

Similarly, we have

The set is defined by .

The energy functional corresponding to (3) can be defined by

Also, we can compute that

To seek the sign-changing solution of (3), we define the set

The main results of this paper are described as follows.

Theorem 1. Assume that hold, then problem (1) possesses a least-energy sign-changing solution such that which changes sign only once.

Theorem 2. Assume that hold. Then problem (1) possesses a solution such that . Moreover,

Theorem 3. Assume that hold. Then problem (2) possesses a sign-changing solution such that , which changes sign only once. Moreover, for any sequence with as, there exists a subsequence of , still denoted by , such that in, where is a sign-changing solution of problem (2) with

Theorem 4. Assume that hold. Then problem (3) has a sign-changing solution such that, which changes sign only once. Moreover, for any sequence withas, there exists a subsequence of , still denoted by, such thatin, whereis a sign-changing solution of (3) with

The rest of the paper is organized as follows. In Section 2, we will give several estimates. In Section 3, some critical lemmas are proved. In Section 4, we will give the proof of the existence of the least-energy sign-changing solution. In section 5, the energy doubling property is proved. Section 6 is devoted to proving the convergence property.

2. Several Estimates

Lemma 5. If the assumptions hold, then

Proof. According to , we can deduce that From (9), (12), (13), and (21), we have The above inequality implies that (20) holds.

Corollary 6. If the assumptions hold and , then From , we have . Therefore, we can immediately get the above conclusion by (20).

Lemma 7. Assume that holds, then We can get (25) by taking in (21).

Lemma 8. If the assumptions hold, then for any , we have We can get the conclusion by a similar deduction as Lemma 5

Corollary 9. If the assumptions hold and , then

3. Some Critical Preliminaries

Lemma 10. If the assumptions hold and with , then there exists a unique pair of positive numbers such that .

Proof. From the definition of the set , implies that Thus, we assume If there is a unique pair of positive numbers such that , then Lemma 10 holds. Next, we will give the detailed proof.
By and , we have that for small enough and for large enough. Thus, there exists such that According to (28) and (29), it is clear that is increasing on for fixed and is increasing on for fixed . Thus, combining (30) we have Miranda’s theorem [20] implies that there exists some point such that , where . Therefore, there exists a positive pair of numbers such that
Next, we will prove that is unique for (28) and (29). Let and be such that From Corollary 6, we have The above two inequalities implies that , that is, . The uniqueness is proved. Therefore, Lemma 10 holds.

Corollary 11. If the assumptions hold and , then there exists a unique such that

Lemma 12. If the assumptions hold, then

Proof. Firstly, by Corollary 6, one has Secondly, for any with , it follows from Lemma 10 that Combining (35) and (36), we can get Lemma 12.

Lemma 13. If the assumptions hold, then can be achieved.

Proof. For all , we have . According to , and the Sobolev embedding theorem, we can get where and are positive constants. Thus we have . Therefore, by (9), (10), (25), and (37), we have Since , thus, for any and . Let be such that . For large , one has Thus, is bounded in for , then there exists such that in . From , we have , that is By a similar deduction as (37), we have for all . From and , for any , there exists such that Thus, . Since is bounded in , there is such that , which implies Choose , then that is . By the compactness of the embedding for , we have Thus By , and the compactness lemma of Strauss [21], we have Also, according to the properties of the solution for the Poisson system, we have By the weak semicontinuity of norm, we have From (40), (44), (46), and (47), we have that is, Since , for all , by (9), (10), (20), and (25), the weak semicontinuity of norm, Fatou’s lemma, and Lemma 12, we can get Thus, Consequently, in and

Corollary 14. Assume that hold. Then and .

Lemma 15 (See for example [3]). Assume that hold. Then there exists a constant and a sequence satisfying

Lemma 16. If the assumptions hold and withthen is a critical point of.

Proof. For , there exist and such that From (23), we have that Let . It follows from (54) that For ,, [22]. Lemma 7 yields a deformation such that (i), if (ii)(iii)From Corollary 6, we have for . For , we know , then it follows from (ii) that
According to (iii) and (54), we have that Thus,
Next, we prove that , which contradicts to the definition of . Let us define and Lemma 10 and the degree theory yields . By (55), we can deduce that on . Consequently, . Therefore, we have for some , so that , which is a contradiction. Thus, (53) does not hold. In other words, is a critical point of , that is, is a sign-changing solution for problem (1).

4. The Existence Result of the Sign-Changing Solutions

In this section, we mainly give the proof of Theorem 17.

Proof of Theorem 17. By Lemma 13 and Lemma 16, there is a such that and . Therefore, is exactly a sign-changing solution of problem (1). Now, we prove that changes sign only once.
We assume by contradiction that , where and .
Let , then , and . Note that and , we have From (9)–(13), (23), (25), (58), and (59), we have Since we have . Therefore, changes sign only once.

5. Energy Doubling Property

Under the above preparations, we give the proof of Theorem 18.

Proof of Theorem 18. By Lemma 15, we know that there exists a sequence satisfying (52), that is, According to (9), (10), (25), and (61), we have for large which shows that is bounded in for . By a standard argument, we can prove that there exists a such that . This suggests that is a nontrivial solution for problem (1) and . On the other hand, by using (9), (10), and (25), the weak semicontinuity of norm and Fatou’s lemma, we have which implies . Thus, . In view of Theorem 1, there exists such that . Therefore, from (11), (24), and Corollary 14, we have which implies the energy of the least-energy sign-changing solution is strictly larger than twice that of the ground state solutions of the Nehari type.

6. The Convergence Property

In this part, we give the convergence property for . Firstly, we have to give some estimates which will be used in the following process. Choose such that , then there exists a constant such that

Thus, by and , there exist and such that

Proof of Theorem 19. It is clear that is allowed in Section 3. Therefore, we can deduce that there exists a such that and , that is, problem (2) has the least-energy sign-changing solution, which changes sign only once.
For , let be a sign-changing solution of (1) obtained in Theorem 1, which changes sign only once and satisfies .
Then, for any , it follows from Lemma 12 and (65)-(66), we have For any sequence with as , by (9), (10), (25), and (67), we have for large Since , is bounded in . Therefore, there exists a subsequence of , still denoted by and , such that in . By a standard argument, we can prove in . Since Thus, , , and . Next, we give the proof of . Choose , from , there exists a such that From Lemma 10, there exists such that , (5.4) implies . Since , then from (9), (10), (15), (16), and (20), we have which implies According to (9), (15), and (72), we have This shows , and the convergence property of is proved.

Proof of Theorem 20. Since the proof is similar as the proof of Theorem 19, we omit the details.

Data Availability

The data used to support the findings of this study are included within the article.

Conflicts of Interest

The authors declares that they have no conflicts of interest.

Acknowledgments

The authors would like to express their sincere gratitude to the anonymous referees for their invaluable comments and suggestions which helped improve the paper greatly. This research was supported by the Program for the Innovative Talents of Higher Education Institutions of Shanxi, the Scientific and Technological Innovation Programs of Higher Education Institutions in Shanxi (201802085), and the innovative research team of North University of China(TD201901), the Fund for Shanxi “1331KIRT.”