Multiple Solutions for a Nonlocal Elliptic Problem Involving -Biharmonic Operator

Abstract

In this paper, using the variational principle, the existence and multiplicity of solutions for -Kirchhoff type problem with Navier boundary conditions are proved. At the same time, the sufficient conditions for the multiplicity of solutions are obtained.

1. Introduction

In this paper, we will discuss the nonlocal elliptic problem involving -biharmonic operator:where is a smooth bounded region. , . , , , and . Furthermore, the function satisfies that is continuous in , for all , and is in , for every, and , for all , where and are partial derivatives of and , respectively. Besides, the functions are continuous and satisfy , for every .

As we all know, the harmonic equation and biharmonic equation in partial differential equation are the most widely used equations in the field of theoretical research and engineering application. Among them, the focus and difficulty in science and engineering is to solve the boundary value problem of biharmonic equation. Therefore, many scholars have carried out a lot of in-depth research on biharmonic problems, see [1–6]. In particular, in [3], when the nonlinear functions F and satisfy certain conditions, Li and Tang studied the -biharmonic problem,by applying the three critical point theorem by Ricceri [7]. Two years later, Masser et al. [8] supposed that the assumption in problem (2) was 0 and imposed appropriate conditions on . Based on the critical point theorem introduced by Bonanno and Molica Bisci, infinitely many solutions were obtained. It well known that the variable exponent case possess more complicated properties than the constant exponent case, and some methods used in the -biharmonic case cannot be applied to the -biharmonic case. Therefore, Allaoui et al. [9] have made a great contribution to such problems, and they continued to extend -biharmonic operator in [8] to -biharmonic case, on the basis of Ricceri’s variational principle [10] and the basic theory of Sobolev space, and the following system is solved:

With the deepening of the investigation, scholars are increasingly paying attention to the solution of nonlocal elliptic equation, see [11–15], for details.

In [16], Ferrara et al. researched the problem:

If the nonlinear term satisfies the certain conditions, then we obtain that there are at least two nontrivial solutions to problem (4) by using the variational method and mountain pass lemma. In the same year, Hssini et al. [17] studied the same problem as in [16] according to Ricceri’s variational principle and obtained multiple solutions of problem (4). In addition, in [18], Xiao and Miao extended problem (4) and continued to study the multiplicity of solutions of -Kirchhoff type problems. In [19], the author extended the conclusion of the problem (4) to the -biharmonic operator.

So far, there are few results on the existence and multiplicity of solutions for -Kirchhoff type problems under Navier boundary condition. Therefore, inspired by the above research, in the present paper, our target is to show the existence and multiplicity of solutions of problem (1), according to the variational principle proposed by Ricceri and the basic results of Lebesgue or Sobolev spaces with a variable exponent.

This article consists of three sections. In Section 2, some basic conclusions and certain essential theorems or propositions of generalized Lebesuge–Sobolev spaces are obtained, which provides the significant framework for the research of variational problems and nonlocal elliptic problems with -biharmonic operator. In Section 3, we have the main results of this article and gain the corresponding proof mainly according to the variational principle by Ricceri. Besides, at the end of Section 3, we give an example involving -Kirchhoff type equations with Ω = [(−1, 1)]², which is actually an application of Theorem 2, that is to say, this example explains our main results very well.

2. Preliminaries

Define the space as follows:which has the norm

The Sobolev space with variable exponents is defined aswhich has the normwhere is a multi-index; moreover, was established.

The closure of in is the . Besides, from [20], we know that and are Banach space and satisfy separability, uniform convexity, and reflexivity.

Proposition 1 (see [20]). Assume; then, and are conjugate spaces and satisfy the Holder inequality:where and .

Denote , which is a separable and reflexive Banach space endowed with the norm:where

From [21], we know that , , and are equivalent norms of .

Proposition 2 (see [9]). Since and is a bounded region, hence, the space to is a compact embedding, and there exists a positive constant such that

Proposition 3 (see [9, 19]). Let , for ; then, we deduce (h1)  (h2)  (h3)  (h4)

Definition 1. If the following equalityholds, for all , then is a weak solution of problem (1).
Define the functional :where .
The functional are well defined and Gateaux differentiable functions for; we haveHence, according to Definition 1, we know that if is the weak solution of problem (1), it is equivalent to that is the critical point of . In addition, since the space to is a compact embedding, so are sequentially weakly lower semicontinuous, and it is clear that is coercive.

Theorem 1 (see [10]). Let X be a reflexive real Banach space. are Gateaux differential functionals; satisfies coercive and sequentially weakly lower semicontinuity and satisfies sequentially weakly upper semicontinuity. If , denotesthen one has(a)For every and every , the functional has a global minimum in , which is a critical point (local minimum) of in X.(b)If , then ; the following alternative holds: either(b1) has a global minimum or(b2) has a series of critical points (local minimum) defined as satisfying .(c)If , then ; the following alternative holds: either(c1) has a global minimum which is a local minimum of or(c2) has a series of pairwise distinct critical points (local minimum) which weakly converges to global minimum of .

3. Main Results and Proof

Theorem 2. Suppose  , .  and two real numbers satisfy . If we putone haswhereThen, for everyProblem (1) has a sequence of solutions such that .

Proof. Since and , if , thenFrom , we know that there exists a sequence satisfying andPutfor all . When , we have .
Sinceaccording to Proposition 3, for , we infer thatThen,Thus,If , we obtain ; then, ; if , we obtain ; then, .
Hence, for large enough ,from (12)thusThen, the inclusion of sets is valid:From (21)–(32), we havePutFrom (22) and (33), we can deduceThus, .
Next, it will prove when ; is unbounded from below. Indeed, on account ofThere exists a column of positive real numbers and such that andNow, we define as follows:Then, andThus,From , we haveWhen is large enough, from (37), (40), and (41), we can obtainsoAccording to Theorem 1 (b), we know that Theorem 2 is established.

Theorem 3. Presume holds and , , , and one haswhereThen, for everyproblem (1) has infinity solutions converging to 0.

Proof. Obviously, .
There is a real sequence satisfying as andPut , for all and ; then, by assumption , we haveHence, .
In the next step, we prove that does not have the local minimum at 0. For , we haveand there exists a column of positive real numbers as and such thatLet be defined by (38); then, we can obtainCombining (41), (51), and (52), we haveObviously, according to Theorem 1 (c), we have completed the proof of Theorem 3. Moreover, is the solution satisfying the conditions and .

Example 1. Let ; for all , and , where a, b, c, and d are positive numbers and . On , are defined asDefine the function bywhere and and is an open unit ball with its center point at .
It is clear that and , for all . The maximum of on is .
Hence,Therefore,where is the measure of .
At the time, , and we deducesothat is to sayThus,; from Theorem 2, we can infer that the equations,have a sequence of unbounded weak solutions.

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Authors’ Contributions

Each part of this paper is the result of the joint efforts of QZ and QM. They contributed equally to the final version of the paper. All the authors have read and approved the final manuscript.

Acknowledgments

This study was supported by the National Natural Science Foundation of China (no. 11861078) and the project of Science Research Fund of Yunnan Education Department (2019J0689 and 2021Y661).