Based on the basic theory and critical point theory of variable exponential Lebesgue Sobolev space, this paper investigates the existence and multiplicity of solutions for a class of nonlocal elliptic equations with Navier boundary value conditions when (AR) condition does not hold and improves or generalizes the original conclusions.

1. Introduction

We deal with the following system:where is a bounded smooth region. The constants and satisfy and , and . The nonlinear term satisfies Carathéodory condition. And, .

The differential equations with p(x)-Laplacian or biharmonic operators are derived from nonlinear elastic mechanics and fluid mechanics. This model can describe the physical phenomenon of “point by point anisotropy.” Various methods have extensively studied the problem of differential operators with variable exponents, as shown in [16]. In [7, 8], the authors used changeable nonlinear terms to outline the boundary of the real image, eliminate the possible noise, and solve the problems in image processing.

Recently, the questionhas been widely concerned by many scholars after Lions [9] found an abstract method to solve this problem. Furthermore, many interesting results can be seen in [1012]. In addition, in [13], Dai and Liu studied the following cases with a variable exponent:According to the variational method and the basis theorem of Sobolev space under suitable assumptions, the infinite positive solutions of the problem with p(x)-Laplace operator is obtained.

As we all know, the Ambrosetti–Rabinowitz [14] (it is written as (AR)) condition can not only ensure that is superlinear with respect to the variable at infinity but also ensure the boundedness of the Palais–Smale (it is written as (PS)) sequence. Therefore, (AR) condition is essential for the study of many boundary value problems, see [1518]. In particular, when a = 1 and b = 0 in problem (1) and (AR) condition holds, the existence of solution of problem (1) is discussed in [16]. In [17], Guo and Zhao made reasonable presupposition on the nonlinear terms and , respectively, and obtained the existence number of solutions for p(x)-Laplace equation in the whole space by applying the basic theory of weighted function and variable exponent space. In [18], Chung, respectively, applied the minimum principle or critical point theory to discuss the problem when the perturbation term is sublinear or superlinear at infinity. In addition, based on the critical point theory, -group index theory, and operator equation theory, Shu, Lai, and Xu [19] constructed a lower semicontinuous convex function with subdifferentiability and obtained infinite subharmonic periodic solutions of second-order neutral nonlinear functional differential equations. There are similar articles such as [20].

In [21], Wang and An studied the following problem:According to the mountain pass lemma satisfying (AR) condition, it is proved that the above elliptic equation has at least two solutions, both of which are nontrivial, where one is positive and the other is negative.

However, although many functions satisfy the superlinear growth condition, the (AR) condition does not hold. For example, in [22], Zhang considered the following problems:When the condition of perturbation term is weaker than (AR), based on the fountain theorem, it is obtained that there are infinite solutions to the above problem.

In [23], the author further investigated the existence of multiple solutions for formula (1) when the (AR) condition is invalid according to the same critical point theory. It should be noted that although problem (1) has been studied in [23], this paper will prove that under weaker conditions, we can still use the fountain theorem to obtain the same result as that in reference [23], which illustrates that our conclusion is more universal. Furthermore, we will obtain mountain pass solutions and infinitely many solutions converging to 0.

This article is mainly composed of four sections. Section 2 introduces the basic theory of Lebesgue and Sobolev spaces and gives the related propositions and theorems needed below. Section 3 first proves that the solution of problem (1) exists under reasonable assumptions and then further proves the existence of at least one nontrivial solution by quoting mountain pass lemma. In the last but most important part (Section 4), the sufficient conditions for the existence of multiple solutions of problem (1) are obtained according to the variational method and critical point theory. It is worth noting that the original results are generalized and improved in this part, and we will explain in detail that our results are better later.

2. Preliminaries


Define the following spaces:having the norm

In addition, the variable exponential Sobolev space is defined as follows:having the normwhere and formula is true. The closure of in is the . Besides, from [24], we know that and are Banach space and satisfy separability, uniform convexity, and reflexivity.

Denote , which is a reflexive and separable Banach space endowed with the normWe haveBy [25], we realize that , , and are equivalent to the norm of .

Define the energy functional as follows:where and . Furthermore, , we know that

Above all, it is easy to obtain that if is the weak solution of problem (1), it means that is the critical point of functional .

Proposition 1 (see [2, 23]). Assume that possesses the cone property; moreover, with then, is a continuous and compact embedding, where

Proposition 2 (see [1]). (1) is a homeomorphism.(2) is a strictly monotone, bounded, and continuous operator.(3) is a mapping of type , namely, on ; if and, furthermore, , then in .

Remark 1. According to Proposition 2, we have that satisfies type and weakly lower semicontinuous.

Proposition 3 (see [24]). Assume ; then, and , we havewhere and are conjugate space.

Proposition 4 (see [2, 24]). Let ; , we deduce

Theorem 1 (see [26]). Let X be a real Banach space. If satisfies (PS) condition andthen has a critical value expressed as with , where .
Owing to as a reflexive and separable Banach space, then there exists such thatThus, . If , it is clear that is the -dimensional subspace of .

Theorem 2 (see [26]). Assume the following:(M1) X is a Banach space, satisfies , and . If for each k = 1, 2, …, there exist such that we have the following:(M2) as (M3) (M4) The functional satisfies Cerami condition (abbreviated as (C) condition), namely, for any such that and as has a convergent subsequence, then has a series of critical values that tend to be positive infinity.

Theorem 3 (see [26]). Assume that (M1) is satisfied and there is such that, for each , there exists such that we have the following:(N1) (N2) (N3) as (N4) For every , satisfies condition, i.e., if any sequence such that , , , and contain a subsequence converging to a critical point of , then has a series of critical values that converge to zero, where these critical values are negativeNext, the nonlinear term always has to verify the following assumptions.
(g0) , we getwhere and .

Remark 2. From Proposition 1, we can infer that X is compact-embedded into and , respectively, that is to say, for any , there are such that and . Besides, let be small enough such that . We have the following:(g1) There exists such that(g2) uniformly in (g3) uniformly in (g4) (g5) , where for a.e. The main conclusions and proofs of the article will be introduced in the following. Note that are positive constants and have different meanings in different places.

3. Existence of Solutions

Theorem 4. Assumewhere ; then, there is a weak solution to formula (1).

Proof. By (22), we deduce that . If is large enough and using Proposition 1, we getThanks to , we deduce thatwhere . Hence, H is coercive. Since X is reflexive and H is weakly lower semicontinuous, then has a minimum point in ; moreover, u is a weak solution of (1).

Lemma 1. If f satisfies (g0) and (g1), then satisfies (C) condition.

Proof. Let be a Cerami sequence, i.e., and . Next, we show that is bounded in. Assume as ; by (g1), there exists such thatPut . By (25), there exists such thatsince and ; so, this is contradictory, which means is bounded. Without loss of generality, if , we have Then, it can be obtained according to conditions (g0), Hölder inequality, and Remark 1, and the specific proof can be seen in [23].

Remark 3. This is the same as the proof of Lemma 1, and it can be obtained that also satisfies the (PS) sequence.

Theorem 5. If satisfies (g0), (g1), (g2), and (g3), then there is a nontrivial solution to problem (1).

Proof. Obviously, H satisfies (PS) sequence and . Next, we need to show (Q2) and (Q3).
First of all, (Q2) is established. From (g0) and (g3), we can infer thatWhen , we havewhere ; therefore, there exist such that for every.
It only remains to prove (Q3). By (g2), we obtain that , there exists that satisfies with and ; we gainwhere and is a normal number. If is sufficiently large and , then . Hence, based on Theorem 1, we can get that has at least one nontrivial solution.

4. Multiple Solutions

The conclusions of this section are as follows.

Theorem 6. If f satisfies (g0), (g1), (g2), and (g4), then equation (1) has a sequence of solutions such that as.

Theorem 7. If f satisfies (g0), (g1), (g3), (g4), and (g5), then equation (1) has a sequence of solutions such that and as .
Next, we will use the fountain theorem and dual fountain theorem to prove Theorem 6 and Theorem 7, respectively.

Lemma 2 (see [2, 18]). If and for any , denote

Lemma 3 (see [2, 18]). If conditions (g0) and (g1) are true, then satisfies the condition.

Proof of Theorem 6. Thanks to Lemma 1 and (g4), it is easy to gain (M1) and (M4). Now, we only need to prove that (M2) and (M3) are tenable.
Let us start with (M2). ; by (g0), we haveand we know and as after thatThen, we show the correctness of (M3). By (29), we infer that for any , with , afterwardsSince and the norm in a finite-dimensional space is equivalent, when is sufficiently large and , we know that .
Therefore, the proof of Theorem 6 is finished.

Remark 4. When (g0), (g2), and (g4) are established, the author of [23] has studied the multiplicity of solutions to equation (1) and assumed thatHowever,Therefore, condition is weaker than condition , which shows that we generalize the original results and our results are better. In other words, our conclusion is more universal.

Proof of Theorem 7. By (g4) and Lemma 3, we will deduce that satisfies (M1) and (N4). Next step, all we have to do is check (N1)–(N3).
The first step is to prove (N1). Due to Lemma 2 and (27), for , with , we deduce thatsince belongs to ; thus, it is clear that That is to say, (N1) was established.
Then, we verify that (N2) is true. and are small enough, by (g5), we haveNow, it only remains to show (N3). Since and , thenFrom (37), , we gainnoting that as ; hence, , i.e., (N3) is satisfied. Consequently, the Theorem 7 is proved.

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.


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