Abstract
In this paper, we study the multiplicity of solutions for a class of quasilinear elliptic equations with pLaplacian 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; pLaplacian; multiple solutions1 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 [1], Shibo Liu considered the existence of multiple nonzero solutions of the Dirichlet boundary value problem
where , denotes the pLaplacian operator, Ω is a bounded domain in with smooth boundary ∂Ω.
Moreover, Aouaoui studied the following quasilinear elliptic equation in [2]:
and proved the multiplicity of solutions of the problem (1.2) by using the nonsmooth critical point theory. One can refer to [3,4] and [5] for more results.
In this paper, we shall investigate the existence of infinitely many solutions of the following problem
where , and , is a given continuous function satisfying
In order to determine weak solutions of (1.3) in a suitable functional space E, we look for critical points of the functional defined by
where . Under reasonable assumptions, the functional J is continuous, but not even locally Lipschitz. However, one can see from [4,6] and [7] that the Gâteauxderivative of J exists in the smooth directions, i.e., it is possible to evaluate
Definition 1.1 A critical point u of the functional J is defined as a function such that , , i.e.,
Our approach to study (1.3) is based on the nonsmooth critical point theory developed in [8] and [9]. 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 [2], 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.
(H_{1}) 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;
(H_{2}) There exist , and such that
(H_{3}) Let a Carathéodory function satisfy , a.e. and
where θ is the same as that in (H_{2}).
(H_{4}) There exists such that
Example 1.1 Let . The following function satisfies hypotheses (H_{1}) and (H_{2})
and the corresponding constants are
Example 1.2 The following function satisfies hypotheses (H_{3}) and (H_{4})
On the other hand, we define the operator . It follows from [12] 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 (H_{1})(H_{4}). 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.
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 [13] and [14]. 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 [2].
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
Our results are based on the techniques of nonsmooth critical point theory. In this section, we recall some basic tools from [8] and [9].
Definition 2.1 Let be a metric space, let be a continuous functional and . We denote by the supremum of the σ’s in such that there exist and a continuous map , satisfying
The extended real number is called the weak slope of I at u.
Note that the notion above was independently introduced in [15], as well.
Definition 2.2 Let be a metric space, let be a continuous functional and . We say that I satisfies , i.e., the PalaisSmale condition at level c, if every sequence in X with and admits a strongly convergent subsequence.
In order to treat the PalaisSmale condition, we need to introduce an auxiliary notion.
Definition 2.3 Let c be a real number. We say that functional I satisfies the concrete PalaisSmale condition at level c ( for short) if every sequence satisfying
possesses a strongly convergent subsequence in E, where is some real number converging to zero.
Remark 2.1 Under assumptions (H_{1})(H_{4}), if the functional J satisfies (1.4), then J is continuous, and for every we have
where denotes the weak slope of J at u.
Remark 2.2 Let c be a real number. If J satisfies , then J satisfies .
Proof Let be a sequence such that
3 Basic lemmas
To derive our main theorem, we need the following lemmas. The first lemma is the version of the AmbrosettiRabinowitz mountain pass lemma [10,11] and [16].
Lemma 3.1LetXbe an infinitedimensional Banach space, and letbe a continuous even functional satisfyingfor every. Assume that
(i) there exist, and a subspaceof finite codimension such that
(ii) for every finitedimensional subspace, there existssuch that
Then there exists a sequenceof critical values ofIwith.
Lemma 3.2Ifis a critical point ofJ, then.
Proof For , , consider the real functions , and defined in ℝ by
and . Denoting and , we can take as a test function in (1.5). Therefore,
From (1.10) and the fact we deduce
Since a.e. in and in E as . It follows from that
Denote . If , then the result is true. In the following discussion, is assumed. By (1.6), we obtain
On the other hand, we have
which implies that
Eventually, one can deduce from (3.2)(3.4) that
By Theorem 5.2 of [17], we get that . Replacing by , we can similarly prove that . We conclude that , and the proof of Lemma 3.2 is completed. □
Lemma 3.3Letbe a bounded sequence inEwith
whereis a sequence of real numbers converging to zero. Then there existssuch thata.e. inand, up to a subsequence, is weakly convergent touinE. Moreover, we have
i.e., uis a critical point ofJ.
Proof Since is bounded in E, and there is a (see [18]) such that, up to a subsequence,
Moreover, since satisfies (3.6), by Theorem 2.1 of [19], we have, up to a further subsequence, a.e. in .
We will use the device of [20]. We consider the test functions
where , and . According to (1.6) and (1.7), we have
Since (3.6) holds by density for every , we can put in (3.6) and obtain that
On the other hand, note that
One can deduce from (3.10) and Fatou’s lemma that
We consider the test functions with , and , , ,
This together with (3.11) can prove that
In a similar way, by considering the test functions , it is possible to prove that
From (3.12) and (3.13), it follows that
Finally, we can deduce (3.7) from (3.14). □
Remark 3.1 (see [21])
Let be a sequence in E satisfying (3.6). Then and
In the following lemma, we will prove the boundedness of a sequence under (1.6), (1.8) and (1.9).
Lemma 3.4Letandbe a sequence inEsatisfying (3.6) and
Proof Calculating , from (3.15) and (3.16), we obtain
From (1.8) and (1.9), it follows that
Moreover, there exist and such that
Therefore, denoting , we obtain from (3.17) that
By virtue of hypothesis (H_{3}), we know that there exist and such that
From (3.18) and (3.19), it follows that
On the other hand, by Hölder’s inequality and Young’s inequality, for all , there exists such that
Using (3.20) and (3.21), we get
Choosing in (3.22), we find that is bounded in E. □
Lemma 3.5Letbe the same as that in Lemma 3.3. Then, up to a subsequence, converges strongly touinE.
Proof By Lemma 3.3, we know that u is a critical point of the functional J. Then, from Lemma 3.2, we get . Therefore, taking as a test function in (3.7), we get
By virtue of is bounded in E, we can assume that there exists satisfying
By Lemma 3.3, a.e. in . Then by Fatou’s lemma, we have
By using (3.23)(3.26) and passing to limit in (3.15), we obtain
On the other hand, by Lebesgue’s dominated convergence theorem and the weak convergence of to u in E, we get
Moreover, since and are bounded in , then we have
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
holds for any , and . Therefore,
According to (1.6), we conclude that converges strongly to u in E. □
Lemma 3.6For every real numberc, the functionalJsatisfies.
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 .
On the other hand, from (1.4), (1.6), (1.9) and (1.10), for , we have
We discuss (4.1) in the following two cases:
In case , by the definition of , we get
i.e., . Therefore, if λ satisfies , there exist small enough and such that
Hence, condition (i) of Lemma 3.1 holds with .
Now we consider a finitedimensional subspace W of E. Let and . From (1.6), we have
By virtue of (1.9) and (1.10), we know that there exist , satisfying a.e. and a positive constant such that
Combining (4.2)(4.3), we have
Since W is finitedimensional, then all norms of W are equivalent. From (4.4), there exists such that
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.
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
We declare that all authors collaborated and dedicated the same amount of time in order to perform this article.
Acknowledgements
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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