In this paper, we study the multiplicity of solutions for a class of quasilinear elliptic equations with p-Laplacian in . In this case, the functional J is not differentiable. Hence, it is difficult to work under the classical framework of the critical point theory. To overcome this difficulty, we use a nonsmooth critical point theory, which provides the existence of critical points for nondifferentiable functionals.
MSC: 35J20, 35J92, 58E05.
Keywords:quasilinear elliptic equations; nondifferentiable functional; p-Laplacian; multiple solutions
1 Introduction and main results
Recently, the multiplicity of solutions for the quasilinear elliptic equations has been studied extensively, and many fruitful results have been obtained. For example, in , Shibo Liu considered the existence of multiple nonzero solutions of the Dirichlet boundary value problem
Moreover, Aouaoui studied the following quasilinear elliptic equation in :
In this paper, we shall investigate the existence of infinitely many solutions of the following problem
where . Under reasonable assumptions, the functional J is continuous, but not even locally Lipschitz. However, one can see from [4,6] and  that the Gâteaux-derivative of J exists in the smooth directions, i.e., it is possible to evaluate
Our approach to study (1.3) is based on the nonsmooth critical point theory developed in  and . Dealing with this class of problems, the main difficulty is that the associated functional is not differentiable in all directions.
The main goal here is to establish multiplicity of results for (1.3), when is odd and is even in s. Such solutions for (1.3) will follow from a version of the symmetric mountain pass theorem due to Ambrosetti and Rabinowitz [10,11]. Compared with problem (1.2) in , problem (1.3) is much more difficult, since the discreteness of the spectrum is not guaranteed. Therefore, we only consider the first eigenvalue .
To state and prove our main result, we consider the following assumptions.
where θ is the same as that in (H2).
and the corresponding constants are
Example 1.2 The following function satisfies hypotheses (H3) and (H4)
On the other hand, we define the operator . It follows from  that the discreteness of the spectrum is not guaranteed. Hence, we only consider the first eigenvalue , where
Next, we can state the main theorem of the paper.
Theorem 1.1Assume thatandsatisfy (H1)-(H4). Moreover, letand, a.e. , . If there exists a positive numberμsuch that, then problem (1.3) has infinitely many distinct solutions in, i.e., there exists a sequence, satisfying (1.3) and, as.
Such a weighted Sobolev space has been used in many previous papers, see  and . Now, we give an important property of the space E, which will play an essential role in proving our main results.
Remark 1.1 One can easily deduce and for . More details can be found in .
Throughout this paper, let denote the norm of E and () means that converges strongly (weakly) in corresponding spaces. ↪ stands for a continuous map, and ↪↪ means a compact embedding map. C denotes any universal positive constant unless specified.
The paper is organized as follows. In Section 2, we introduce the nonsmooth critical framework and preliminaries to our work. In Section 3, we give some lemmas to prove the main result. Finally, the proof of Theorem 1.1 is presented in Section 4.
2 Nonsmooth critical framework and preliminaries
Note that the notion above was independently introduced in , as well.
Definition 2.2 Let be a metric space, let be a continuous functional and . We say that I satisfies , i.e., the Palais-Smale condition at level c, if every sequence in X with and admits a strongly convergent subsequence.
In order to treat the Palais-Smale condition, we need to introduce an auxiliary notion.
3 Basic lemmas
On the other hand, we have
which implies that
Eventually, one can deduce from (3.2)-(3.4) that
By Theorem 5.2 of , we get that . Replacing by , we can similarly prove that . We conclude that , and the proof of Lemma 3.2 is completed. □
i.e., uis a critical point ofJ.
Proof Since is bounded in E, and there is a (see ) such that, up to a subsequence,
Moreover, since satisfies (3.6), by Theorem 2.1 of , we have, up to a further subsequence, a.e. in .
We will use the device of . We consider the test functions
On the other hand, note that
One can deduce from (3.10) and Fatou’s lemma that
This together with (3.11) can prove that
From (3.12) and (3.13), it follows that
Finally, we can deduce (3.7) from (3.14). □
Remark 3.1 (see )
From (1.8) and (1.9), it follows that
From (3.18) and (3.19), it follows that
Using (3.20) and (3.21), we get
By using (3.23)-(3.26) and passing to limit in (3.15), we obtain
Therefore, from the definition of weak convergence, we obtain
Combining (3.27)-(3.32), it follows that
It is well known that the following inequality
4 Proof of Theorem 1.1
We discuss (4.1) in the following two cases:
Combining (4.2)-(4.3), we have
The authors declare that they have no competing interests.
We declare that all authors collaborated and dedicated the same amount of time in order to perform this article.
The authors express their sincere thanks to the referees for their valuable criticism of the manuscript and for helpful suggestions. This work has been supported by the Natural Science Foundation of China (No. 11171220) and Shanghai Leading Academic Discipline Project (XTKX2012).
Liu, SB: Multiplicity results for coercive p-Laplacian equations. J. Math. Anal. Appl.. 316, 229–236 (2006). Publisher Full Text
Aouaoui, S: Multiplicity of solutions for quasilinear elliptic equations in . J. Math. Anal. Appl.. 370(2), 639–648 (2010). Publisher Full Text
Alves, CO, Carrião, PC, Miyagaki, OH: Existence and multiplicity results for a class of resonant quasilinear elliptic problems on . Nonlinear Anal.. 39, 99–110 (2000). Publisher Full Text
Arcoya, D, Boccardo, L: Critical points for multiple integrals of the calculus of variations. Arch. Ration. Mech. Anal.. 134, 249–274 (1996). Publisher Full Text
Arcoya, D, Boccardo, L: Some remarks on critical point theory for nondifferentiable functionals. NoDEA Nonlinear Differ. Equ. Appl.. 6, 79–100 (1999). Publisher Full Text
Ambrosetti, A, Rabinowitz, PH: Dual variational methods in critical point theory and applications. J. Funct. Anal.. 14, 349–381 (1973). Publisher Full Text
Brasco, L, Franzina, G: On the Hong-Krahn-Szego inequality for the p-Laplace operator. Manuscr. Math.. 141, 537–557 (2013). Publisher Full Text
Bartsh, T, Wang, ZQ: Existence and multiplicity results for some superlinear elliptic problems on . Commun. Partial Differ. Equ.. 20, 1725–1741 (1995). Publisher Full Text
Rabinowitz, PH: On a class of nonlinear Schrödinger equations. Z. Angew. Math. Phys.. 43, 270–291 (1992). Publisher Full Text
Boccardo, L, Murat, F: Almost everywhere convergence of the gradients of solutions to elliptic and parabolic equations. Nonlinear Anal.. 19, 581–597 (1992). Publisher Full Text