Abstract
In the paper, we study the existence of weak solutions for a class of new nonlocal problems involving a -Laplacian operator. By using Ekeland’s variational principle and mountain pass theorem, we prove that the new -Kirchhoff problem has at least two nontrivial weak solutions.
1. Introduction and Main Result
In this paper, we consider the following nonlocal -Kirchhoff problem:
where is a smooth bounded domain with boundary , are constants, with , is called -Laplacian operator. is a real parameter, for , .
The study of variational problems with nonstandard -growth conditions has been a new and interesting topic. These problems arise from the image processing model and stationary thermorheological viscous flows; we can refer to [1, 2]. Problem (1) is related to the stationary problem
where , , , , are constants which represent some physical meanings; is Young’s modulus; is the lateral displacement; and , are the external forces. When , it extends the classical D’Alembert wave equation for free vibrations of elastic strings. Since problem (2) is no longer a pointwise identity, it is often called a nonlocal problem. In recent years, nonlocal elliptic problems have attracted wide attention, and some important and interesting results have been established (see [3–5]).
In the past few decades, many people studied the following -Kirchhoff problem: where be a bounded open subset of with a -boundary , is a bounded below function, is a Carathéodory function. For example, Dai in [4] using the three-critical-point theorem obtained the existence of solutions for problem (3). Moreover, when , Zhou and Ge in [6] studied the existence of the solution for the nonlocal problem (3) by using the fibering map approach for the corresponding Nehari manifold. In recent years, some people are starting to pay attention to the case in problem (3). Obviously, is not bounded below. Therefore, this is a new class of nonlocal problems. In fact, for the case , some results are given in [7–9]. However, few literatures have considered this new nonlocal -Kirchhoff problem. Recently, the authors in [10] have considered the following -Kirchhoff-type problem: where are constants, is a bounded smooth domain, with , is a real parameter, and is a continuous function. Under appropriate hypotheses, the author used a mountain pass theorem and fountain theorem to obtain the existence and multiplicity of nontrivial solutions for problem (4).
Inspired by the above facts and aforementioned papers, the main purpose of this paper is to study the existence of two nontrivial solutions for problem (1). Before stating our main results, we make some assumptions on the functions , , , and .
(H1) , , with ,
(H2) and
(H3)
Now, our main result is as follows:
Theorem 1. Assume that the function , , , , satisfies . If (H1)–(H3) hold, then there exists such that problem (1) has at least two nontrivial weak solutions for any .
2. Preliminaries
In order to discuss problem (1), we need some theories on spaces and which we will call generalized Lebesgue-Sobolev spaces. For more details on the basic properties of these spaces, we refer the readers to Fan and Zhao [11]. Set ; denoted the set of all measurable real functions defined on .
For any , the variable exponent Lebesgue space is defined as
with the norm
The variable exponent Sobolev space is defined as with the norm
Define as the closure of in . From [11], we know that the spaces and are all separable and reflexive Banach spaces.
Moreover, there is a constant , such that for all . Therefore, and are equivalent norms on . In addition, we can recall the following properties of the variable exponent spaces.
Lemma 2 (see [11]). For any , with , the inequality holds as follows: Moreover, if , , and , then for any , , and , the following inequality holds: Let us now recall the modular function which plays an important role in the variable order Lebesgue spaces and which is defined by
Lemma 3 (see [10]). For any , the following properties hold: (i)(ii)(iii),
Lemma 4 (see [10]). For , such that for all , there is a continuous embedding If we replace with , the embedding is compact.
Lemma 5 (see [12]). Assume that , . If , then we have
Lemma 6 (see [10, 13]). Let , for all , then , and for all , , and the following properties hold: (i) is convex and sequentially weakly lower semicontinuous(ii) is a mapping of type ; that is, if in and , then in (iii) is a strictly monotone operator and homeomorphism
Definition 7. We say that is a weak solution of problem (1), if for any , it satisfies the following: The energy functional associated to problem (1) is defined as Obviously, is a functional and for all . It is well known that the weak solution of (1) corresponds to the critical point of the functional on .
Definition 8. Let be a Banach space and .
We say that satisfies the for , if any sequence satisfying and in as has a convergent subsequence.
3. Proof of Main Result
In this section, the existence of nontrivial solutions for problem (1) is obtained by using a mountain pass lemma combined with Ekeland’s variational principle.
Lemma 9. Assume that the conditions of Theorem 1 hold. Then, the function satisfies the condition with for small enough.
Proof. Firstly, we prove that is bounded in . Let be a sequence such that . Arguing by contradiction, we assume that .
According to (H2), we have for all , where , by Lemmas 3–5, we have
Dividing the above inequality by , passing to the limit as , note that and , we obtain a contradiction with being sufficiently small. It follows that is bounded in .
Secondly, we will prove that has a convergent subsequence in .
Since is bounded, there exists a subsequence, still denoted by and such that
In view of (19) and Lemma 2, we have
as , where and . Similarly, we also have as .
Thus,
According to , we have . Therefore,
as . So, we can deduce from (21) that
Since is bounded in , passing to a subsequence, if necessary, we assume that
If , then . For any , by (19) and the Hölder inequality, it implies that
Since
and , we have as for fix . Therefore, .
By the fundamental lemma of the variational method, we obtain that
It follows from and that . So
Hence, for , we see that
which contradicts the fact that . Then, is not true. Hence,
By (23), we obtain that
Invoking the condition (see Lemma 6), we can deduce that strongly in as . So, satisfies the condition. The proof is complete.☐
Lemma 10. Assume that the conditions of Theorem 1 hold; then, there exist , , such that for any and with .
Proof. Let and with . By Hölder inequality and Lemmas 3–5, we can deduce that Since , we can choose sufficiently small such that . Set We can deduce that for any , there exists such that for any with , we have .□
Lemma 11. Assume that the conditions of Theorem 1 hold; then, there exists with such that .
Proof. Choosing satisfies and . We have Since , we obtain as . Then, for large enough, we can take so that and .□
Lemma 12. Assume that the conditions of Theorem 1 hold; there exists such that , , and for all small enough.
Proof. Letting with , for all small enough, we get the estimate Then, for any , with , we conclude that .□
Proof of Theorem 1. By Lemma 10, we have , where . And from Lemma 12, there exists such that for small enough. As in the proof of Lemma 10, for any and , it follows that
Note that for , we have
By applying Ekeland’s principle in (see [14–16]), we obtain that there exists a sequence of . It follows from and Lemma 9 that satisfies the condition. Therefore, one has a subsequence still denoted by and such that in and , , which implies that is a solution of problem (1).
From Lemmas 10 and 11, we see that the functional has the mountain pass geometry. Define
Noting that , for all , we have
So, one has . Letting be a sequence of , by Lemma 9, we obtain which satisfies the condition.
By the mountain pass theorem (see [10, 17]), we obtain that problem (1) has the second solution with . Noting that , we know that and are two nontrivial solutions of problem (1).
Data Availability
The findings in this research do not make use of data.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Acknowledgments
This paper is supported by the National Natural Science Foundation of China (No. 11861021, No. 11861052).