Infinitely many solutions for a class of quasilinear elliptic equations with p-Laplacian in
© Jia et al.; licensee Springer 2013
Received: 15 December 2012
Accepted: 22 July 2013
Published: 6 August 2013
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.
1 Introduction and main results
where , denotes the p-Laplacian operator, Ω is a bounded domain in with smooth boundary ∂ Ω.
for all and .
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.
Suppose that and .
(H1) Let be a function such that
for each , is measurable with respect to x;
for a.e. , is a function of class with respect to s;
there exist such that(1.6)(1.7)
where θ is the same as that in (H2).
where is a positive constant.
Next, we can state the main theorem of the paper.
Theorem 1.1 Assume that and satisfy (H1)-(H4). Moreover, let and , a.e. , . If there exists a positive number μ such that , then problem (1.3) has infinitely many distinct solutions in , i.e., there exists a sequence , satisfying (1.3) and , as .
To explain our result, we introduce some functional spaces. We define the reflexive Banach space E of all functions with the norm .
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
The extended real number is called the weak slope of I at u.
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.
possesses a strongly convergent subsequence in E, where is some real number converging to zero.
where denotes the weak slope of J at u.
Remark 2.2 Let c be a real number. If J satisfies , then J satisfies .
By Remark 2.1, we have . Taking , the conclusion follows. □
3 Basic lemmas
- (i)there exist , and a subspace of finite codimension such that
- (ii)for every finite-dimensional subspace , there exists such that
Then there exists a sequence of critical values of I with .
Lemma 3.2 If is a critical point of J, then .
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., u is a critical point of J.
Moreover, since satisfies (3.6), by Theorem 2.1 of , we have, up to a further subsequence, a.e. in .
Finally, we can deduce (3.7) from (3.14). □
Remark 3.1 (see )
In the following lemma, we will prove the boundedness of a sequence under (1.6), (1.8) and (1.9).
Then is bounded in E.
Choosing in (3.22), we find that is bounded in E. □
Lemma 3.5 Let be the same as that in Lemma 3.3. Then , up to a subsequence, converges strongly to u in E.
According to (1.6), we conclude that converges strongly to u in E. □
Lemma 3.6 For every real number c, the functional J satisfies .
Proof Let be a sequence in E satisfying (3.6) and (3.16). By Lemma 3.4, is bounded in E. Therefore, the conclusion can be deduced from Lemma 3.5. □
4 Proof of Theorem 1.1
It is easy to check that the functional J is continuous and even. Moreover, by Remark 2.2 and Lemma 3.6, J satisfies for every .
We discuss (4.1) in the following two cases:
Hence, condition (i) of Lemma 3.1 holds with .
In view of , we deduce that the set is bounded in E and condition (ii) of Lemma 3.1 holds. By Lemma 3.1, the conclusion follows.
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).
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