Existence and Uniqueness of Positive Solution for -Laplacian Kirchhoff-Schrödinger-Type Equation

Abstract

We study the existence and uniqueness of positive solution for the following -Laplacian-Kirchhoff-Schrödinger-type equation: , where , are parameters, and are under some suitable assumptions. For the purpose of overcoming the difficulty caused by the appearance of the Schrödinger term and general singularity, we use the variational method and some mathematical skills to obtain the existence and uniqueness of the solution to this problem.

1. Introduction and Main Results

In this paper, we discuss the following problem:where is a smooth bounded domain with boundary , are parameters, is the -Laplacian operator, and , with .

In recent years, a lot of scholars have studied the singular Kirchhoff problem (for more details, we refer the reader to [1–4]), the Schrödinger-Poisson system (we refer the reader to [5–8]), and the Kirchhoff-Schrödinger-Poisson system (we refer the reader to [9–12]). The authors use various methods to obtain the properties of the solution, which makes such problems very interesting. Inspired by the above papers, later scholars begin to make some expanding study about the above problems. For example, in [13], Guo and Nie studied the existence and multiplicity of nontrivial solutions for -Laplacian Schrödinger-Kirchhoff-type equations by variational methods. For a more complex situation, we refer the reader to [14]. The related studies on the elliptic equations also can be found in [15–26].

However, up to now, no paper has appeared in the literature which discusses the existence and uniqueness of the positive solution for the -Laplacian-Kirchhoff-Schrödinger-type problem. This paper attempts to fill this gap in the literature. Inspired by the above works, in this paper, we try to study the existence and uniqueness of solution to the problem (1) by using the variational method.

Next, we will make some assumptions about and .

satisfies that there exists , such that is nonincreasing on , , and there exists such that

satisfies ,  a.e.  .

and there exists a constant , such that

and the minimum of can be achieved in . In other words, there exists a constant , such that .

   is bounded in satisfies .

There exists a constant such thatwhere denotes the maximal value in .

In this paper, we will make full use of the following definitions.

First, we define the space and the norm

We denote the norm in by .

By and the Poincaré inequality, we can deduce that the embedding is continuous. Thus, according to the continuity of the embedding , there are constants such that

We make further assumptions for convenience. We assume for all . Since in , we know there exists , such thatwhich implies

Also, from the fact that , we can get that is continuous on . Thus for any , by the conditions , (8), (6) and Hölder inequality, we havewhere for all and are some positive constants. Next, we can define the energy functional corresponding to problem (1):By a simple computation, we can get

It is clear that with is called a weak solution of the problem (1) if for any it holdsFinally, we will give the main results of the paper.

Theorem 1. If with and the assumptions and hold, then the problem (1) possesses a positive solution for any . Moreover, this solution is a global minimizer of .

Theorem 2. If and the assumptions and hold. Moreover, assume that is nondecreasing on , then the solution for problem (1) is unique for any .

Remark 3. The result obtained in the paper is an expanding study of the Kirchhoff-Schrödinger-type equation ; the difficulty is posed by the degenerate quasilinear elliptic operator. We mainly use the variational method to solve the problem.

This paper is organized as follows. In Section 2, we will give a preliminary. In Section 3, we will prove the main results.

In this paper, denote various positive constants, which may vary from line to line.

2. Preliminary

To prove the main results in this paper, we will employ the following important lemma.

Lemma 4. If the assumptions , , , and hold, then attains the global minimum in ; that is, there exists such that , and .

Proof. For any , by (5), (9)–(11), we can getSince and , we can obtain that is coercive and bounded from below on . The definition that makes some sense.
Since in the condition , there exists such thatChoosing a nonnegative function with , then for any , by (5), (10), (11), (15), we haveSince and , we can get that for small enough. That is .
On the basis of the definition of , we can deduce that there exists a sequence such that . Since is coercive and , is bounded in . Going if necessary to a subsequence, still denoted by , there exists such thatas . It follows from (8) and Sobolev embedding theorem that is bounded in . Moreover, from the continuity of , we can get that . Thus, we obtain in .
Since , we haveMoreover, by Fatou’s lemma, we haveAccording to the weakly lower semicontinuity of the norm, (18) and (19), we havewhich yields . The proof is completed.

3. Proof of Main Results

Proof of Theorem 1. Since , then . Thus we may assume . Owing to , we know . Next we will give the two-step proof.
(i) Firstly, we shall prove .
For any with a.e.   and , we haveLetting , we can getThus, by Fatou’s lemma and Lemma   in [27], we can getLet be the first eigenfunction of the operator with the Dirichlet boundary and for all . Taking in (23), by and the condition , we havewhich implies a.e. by the condition . If not, there exists such that and for all . Then by Lemma   in [27], we can getIt is a contradiction. So the claim is true.
(ii) is exactly a solution of the problem (1); that is, satisfies (13):To obtain the conclusion, we define a function ; that is,From the above discussion, we know attains its minimum at . By Lemma   in [27], we can get that is differentiable at and ; that is,For each and , we define andThen and
Inserting into (23) and using (28), we can get thatwhich implies thatNext we define for all . By simple computation, we can deduce that is a nonincreasing sequence of measurable sets and Thus we haveLet ; then and as Selecting in (31), we haveAccording to the arbitrariness of , this inequality also holds for . Combining (33), we can get that, for any ,Thus, is exactly a weak solution of the problem (1). By Lemma 4, we know . Therefore, is exactly a global minimizer solution.

Proof of Theorem 2. Assume that is also a solution of problem (1). Letting , according to the definition of the weak solution and (26), we can getwhich impliesNext, we will make some estimates for the equation.
(i) In fact, we estimate as follows.(ii)(iii) Since is nondecreasing on , we have Next, we estimate the left side of (37), according to the conditions and , we can prove thatThus by a simple deduction and , one hasIt follows from (37), (38), (40)–(42), and (44) and we havewhere . Since in the condition , we can get that , that is, . Therefore, the solution of the problem (1) is unique.

Data Availability

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

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

This research was supported by the Distinguished Youth Science Foundation of Shanxi Province (2015021001), the Outstanding Youth Foundation of North University of China (no. JQ201604), and the Youth Academic Leaders Support Program of North University of China.