In this note, we study the existence and multiplicity of solutions for the quasilinear elliptic problem as follows:
where is a bounded domain with a smooth boundary. The existence and multiplicity of solutions are obtained by a version of the symmetric mountain pass theorem.
Keywords:Orlicz-Sobolev spaces; symmetric mountain pass theorem; quasilinear elliptic equations
In this note, we discuss the existence and multiplicity of solutions of the following boundary value problem:
where is a bounded domain with a smooth boundary ∂Ω. The function a is such that defined by
is an increasing homeomorphism from R onto itself and the continuous function satisfies , . Especially, when , the problem (1.1) is the well-known p-Laplacian equation. There is a large number of papers on the existence of solutions for the p-Laplacian equation. But the problem (1.1) possesses more complicated nonlinearities. For example, it is inhomogeneous and has an important physical background, e.g.,
(a) nonlinear elasticity: , ;
(b) plasticity: , , ;
(c) generalized Newtonian fluids: , , .
So, in the discussions, some special techniques are needed, and the problem (1.1) has been studied in an Orlicz-Sobolev space and received considerable attention in recent years; see, for instance, the papers [1-9]. In paper , Fang and Tan discussed the problem (1.1) under the conditions that was odd in t. They got the first result that when , and for , , the problem (1.1) had a sequence of solutions by genus theory. The second result is that when satisfies , , , and as , the problem (1.1) has infinitely many pairs of solutions which correspond to the positive critical values by the symmetric mountain pass theorem.
Motivated by their results, in this note, we discuss the problem (1.1) when is still odd in t but it satisfies weaker conditions than ; and furthermore, we need not know the behaviors of near the zero. If , we can get multiplicity of solutions by a version of the symmetric mountain pass theorem.
The paper is organized as follows. In Section 2, we present some preliminary knowledge on the Orlicz-Sobolev spaces and give the main result. In Section 3, we make the proof.
Obviously, the problem (1.1) allows a nonhomogeneous function p in the differential operator defining the problem (1.1). To deal with this situation, we introduce an Orlicz-Sobolev space setting for the problem (1.1) as follows.
then P and are complementary N-functions (see ), which define the Orlicz spaces and respectively.
Throughout this paper, we assume the following condition on P:
Under the condition (p), the Orlicz space coincides with the set (equivalence classes) of measurable functions such that
and is equipped with the (Luxemburg) norm, i.e.,
We will denote by the corresponding Orlicz-Sobolev space with the norm
and define as the closure of in . In this note, we will use the following equivalent norm on :
Now, we introduce the Orlicz-Sobolev conjugate of P, which is given by
where we suppose that
Let , . Throughout this paper, we assume that . Now, we will make the following assumptions on .
( ) There exists an odd increasing homeomorphism h from R to R, and nonnegative constants , such that
and , , where
then we can obtain complementary N-functions which define corresponding Orlicz spaces and .
Similar to the condition (p), we also assume the following condition on H:
In order to prove our results, we now state some useful lemmas.
Under the condition (p), the spaces , and are separable and reflexive Banach spaces.
Under the condition ( ), the embedding is compact.
Let , we have
(1) if , then ;
(2) if , then ;
(3) if , then ;
(4) if , then .
Let , whereEis a real Banach space andVis finite dimensional. Suppose is an even functional satisfying and
( ) there is a constant such that ;
( ) there is a subspaceWofEwith and there is such that ;
( ) considering given by ( ), Isatisfies (PS)cfor .
ThenIpossesses at least pairs of nontrivial critical points.
Using the version of the symmetric mountain pass theorem mentioned above, we can state our result as follows.
Theorem 2.1Assume that is odd int, satisfies ( ) with and the following assumptions:
( ) there exist and , and , , such that for every , a.e. in Ω.
( ) there is with such that uniformly a.e. in .
Then for any given , the problem (1.1) possesses at leastkpairs of nontrivial solutions.
3 Main results and proofs
In this section, we assume that and , is called a weak solution of the problem (1.1) if
and we know that the critical points of I are just the weak solutions of the problem (1.1).
For E is a separable and reflexive Banach space, then there exist (see ) and such that
Now, we set , , so
Lemma 3.1Given , there is such that for all , .
Proof We prove the lemma by contradiction. Suppose that there exist and for every such that . Taking , we have for every and . Hence, is a bounded sequence, and we may suppose, without loss of generality, that in . Furthermore, for every since for all . This shows that . On the other hand, by the compactness of embedding , we conclude that . This proves the lemma. □
Lemma 3.2Supposefsatisfies ( ), then there exist and such that
Proof Now suppose that . From ( ), we know that
Consequently, considering to be chosen posteriorly by Lemma 3.1, we have for all and j sufficiently large,
Now, taking and noting that , if , we can choose such that , , and for every , , the proof is complete. □
Lemma 3.3Supposefsatisfies ( ). Then given , there exist a subspaceWof and a constant such that and .
Proof Let and be such that , and . First, we take with . Considering , we have . Let and be such that , and . Next, we take with . After a finite number of steps, we get such that , , and for all . Let , by construction, , and for every .
Since , if , then . Now, it suffices to verify that
From the condition ( ), given , there is such that for every , a.e. x in ,
Consequently, for and ,
where and . Observing that W is finite dimensional and we have , , the inequality is obtained by taking ; the proof is complete. □
Lemma 3.4Supposefsatisfies ( ), thenIsatisfies the (PS) condition.
Proof We suppose that ,
Noting that , , is bounded. By , Lemma 3.1, we know that I satisfies the (PS) condition. □
Proof of Theorem 2.1 First, we recall that , where and are defined in (3.1). Invoking Lemma 3.2, we find , and I satisfies with . Now, by Lemma 3.3, there is a subspace W of with and such that I satisfies ( ). Since and I is even, we may apply Lemma 2.4 to conclude that I possesses at least k pairs of nontrivial critical points. The proof is complete. □
The authors declare that they have no competing interests.
All authors read and approved the final manuscript.
Project supported by Natural Science Foundation of China, Tian Yuan Special Foundation (No. 11226116), Natural Science Foundation of Jiangsu Province of China for Young Scholar (No. BK201209), the China Scholarship Council, the Fundamental Research Funds for the Central Universities (No. JUSRP11118) and Foundation for young teachers of Jiangnan University (No. 2008LQN008).
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