Abstract
In this article, we consider a class of nonlocal p(x)-Laplace equations with nonlinear boundary conditions. When the nonlinear boundary involves critical exponents, using the concentration compactness principle, mountain pass lemma, and fountain theorem, we can prove the existence and multiplicity of solutions.
1. Introduction
In this article, we study the following problem:where is a bounded domain with smooth boundary, is the outer unit normal derivative, is the p(x)-Laplace operator, and is a continuous function on , .
There are many relevant conclusions about the study of p-Laplace equations with critical exponentials (see [1–3] and references therein). In [1], the authors studied the following problem:where with . Under several conditions on and , the authors proved the existence of infinitely solutions of problem (2). In (2), When the function , , the relevant results were obtained in [2].
In [4, 5], the general operator (p, q)-Laplacian was considered and also concentration results were produced, while in [6], the existence in bounded sets was proved for a p-Laplacian Dirichlet problem via blowup technique. In [7], the generalized critical Schrödinger equations were considered.
As we know, the Lions concentration compactness principle (see [8]) is a basic tool to prove the existence of solutions when handling nonlinear elliptic equations with critical growth. In [9, 10], the authors extended the Lions concentration compactness principle to the variable exponent. In [11–13], by applying the concentration compactness principle (see [9, 10]), the existence of solutions to the p(x)-Laplace equation with Dirichlet boundary conditions were studied.
In [14], the following problem,was discussed, where relates to the critical exponent. The authors proved that there are infinitely many small solutions to this problem using the concentration compactness principle (see [5]) and the symmetric mountain pass theorem (see [15]).
With the further study of the problem, Kirchhoff-type equations (also known as nonlocal problems) have also attracted extensive attention from scholars (see [16–19]). In [18], according to the variational method and the mapping theorem, he obtained some conclusions on the existence and multiplicity of the problem under weaker assumptions.
However, there are few conclusions for Kirchhoff-type equations with critical growth conditions and nonlinear boundary conditions. Therefore, inspired by the above research, this paper discusses the problem in (1). The main results of this article are the following.
Theorem 1. Suppose and are continuous functions which satisfy the following conditions: , , such that , ; , , such that , , where ; , such that ; , , such that , , where ; , , such that , ; , such that ; satisfies the Caratheodory condition, and there exists , such that where , , , , such that , , ; , such that for and uniformly in ; , such that , , where ; , ; , such that for and uniformly; , , such that , , .
When the conditions , , and are satisfying, equation (1) has a nontrivial solution.
Theorem 2. Under the condition that Theorem 1 holds, the following hypotheses are also satisfied: when , , we have when , , we have
Then, we obtain infinitely many solutions to equation (1), and as , where , , , and denote different positive constants.
2. Preliminaries
In this section, we give some properties and definitions of and to deal with equation (1).
Let be a bounded region, and let
We can introduce the norm on bywhich is a Banach space.
The definition of space is as follows:if the following norm is introduced:
It is well known that is also a Banach space. Specifically, its dual space is , where . For every and , we have
By virtue of inequality holds (see [20, 21]).
Proposition 3 (see [20, 21]). Let , ; then, we have(1)(2); (3)
Proposition 4 (see [20, 21]). (1) is a reflexive, separable Banach space(2)If , then the embedding from to is continuous and compact
Proposition 5 (see [22]). Let be an open bounded region with a Lipschitz boundary.
Assume that , , and that satisfies the condition.
Then, the boundary trace embedding from to is compact, with is the embedding constant.
In this paper, we denote , , and we let “” and “” represent weak convergence and strong convergence, respectively.
Below, we give the definition of weak solutions for equation (1).
Definition 6. A function is a weak solution of equation (1), if, for any ,where and is the surface measure on .
Functional in associated to the equation in equation (1):where .
We define an operator by
Definition 7 (see [14]). If any sequence , which satisfies that is bounded and as , has a convergent subsequence, then is said to satisfy the Palais–Smale condition ((PS) condition for short).
Theorem 8 (see [23]). Assume that is a Banach space; if is said to satisfy the (PS) condition and . Suppose
Then, has a critical value.where
Let be a separable, reflexive Banach space; then, , , and we have
For , we have
Theorem 9 (see [24]). Let , . If, for every , there exists , such that satisfies the (PS) condition for every Then, has an unbounded sequence of critical values.
3. Local (PS) Condition
Lemma 10. Suppose that functions and are continuous which satisfy the conditions: , , and satisfy the conditions , and hold. Then, all (PS) sequences of are bounded in .
According to the conditions of Lemma 10, we can know that the nonlinear boundary of (1) involves critical exponents and, thus, the inclusion from to loses compactness; we can no longer expect the (PS) condition to hold. However, we can solve this difficulty by using the concentration compactness principle.
We use the following lemma to prove that satisfies the local (PS) condition:
Lemma 11 (see [10]). Suppose that and are two continuous functions, such that
Let in , such that
Note that is a measure supported on . Assume that . Then, for some countable index, set , we havewhere . In the Sobolev trace embedding theorem, is the best constant.
Proof of Lemma 10. Let and denoteLet us make the following assumptions.
and a function satisfyingsuch that satisfies For in , there exists small enough, such that satisfiesFor convenience, we defineFor a large enough , according to , we haveUnder assumptions , we obtain .
The conditions , imply that , , when is large enough,Let be a (PS) sequence and assume .
Since , we haveAccording to the Young inequality, we obtainandAccording to the embedding theorem (see [19, 20]), it follows thatIt is not hard to see thatSubstitute equations (30)–(33) into the above equation; then,When the positive constant is small enough, we haveTherefore,Because , is bounded in .
Theorem 12. Let be a (PS) sequence, with energy level c. If , then there exists a subsequence in .
Proof. According to Lemma 10, if is a PS sequence, it can be concluded that is bound in . According to Lemma 11, we know that there exists a subsequence (still denoted as ), such thatLet , and define , such thatConsider . As in , we obtaini.e.,According to the inequality, we haveIt is easy to verify thatHence, from equations (37)–(43), we haveThen,When , we conclude that . Then, through equation (39), we obtain , which suggests thatSuppose that the first case is true; for some ,This is not true. Consequently, for every . Furthermore, when , we haveWe have that is bounded. Then, for a subsequence and , we have in . Observe thatIn fact, it is clear thatUsing the inequality and the fact , , we obtainBecause in , according to Proposition 5, we obtain that is compactly embedded . Thus, we obtainThrough equations (51)–(53), we can deduce thatIt is known thatCombining equations (54) and (55), we can deduce thatThus, according to Proposition 3 (3), we can prove that , .
4. The Proof of Main Results
Proof of Theorem 1. We use the mountain pass theorem to find critical values below level ; thus, we need to verify that the functional satisfies Theorem 8.
According to Lemma 10, the function satisfies the local (PS) condition. Apparently, .
First, we verify . If is small enough, thenWe define . Because and , we can easily obtain that for some sufficiently small.
Next, we verify . For sufficiently large , from it follows that ; through , , we get that ; implies that ; and imply that .
Next, we fix , and then we obtainFor large enough, let ; because , as .
We can draw the subsequent results from the Fountain theorem, which is similar to the proof of Theorem 4.8 in [25].
Proof of Theorem 2. We prove the result using Theorem 9. From and , it can be known that the functional is an even energy functional and satisfies the local (PS) condition.
We assume that the ; thus,We obtain the function below if .Now, we take ; accordingly, we haveBecause , , and .
For the other cases, using a similar method, we obtain, since , , . Thus, is true.
According to and , we obtainLet ; then, we haveBecause , all norms on are equivalent. Therefore,When , we have and . Thus, is true.
On the basis of the proofs of and , we let . Then, the conclusion is valid.
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 LJ and MQ. They contributed equally to the final version of the paper. All the authors have read and approved the final manuscript.
Acknowledgments
This project was supported by the National Natural Science Foundation of China (Grant no. 11861078).