A note on the existence of solutions for a class of quasilinear elliptic equations: an Orlicz-Sobolev space setting

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.

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 [9], 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 [9]; 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.

Let

then P and are complementary N-functions (see [10]), 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

Let

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.

Lemma 2.1[10]

Under the condition (p), the spaces, andare separable and reflexive Banach spaces.

Lemma 2.2[10]

Under the condition (), the embeddingis compact.

Lemma 2.3[2]

Let, we have

(1) if, then;

(2) if, then;

(3) if, then;

(4) if, then.

Lemma 2.4[11-13]

Let, whereEis a real Banach space andVis finite dimensional. Supposeis an even functional satisfyingand

() there is a constantsuch that;

() there is a subspaceWofEwithand there issuch that;

() consideringgiven by (), Isatisfies (PS)_cfor.

ThenIpossesses at leastpairs 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 thatis odd int, satisfies () withand the following assumptions:

() there existand, and, , such thatfor every, a.e. in Ω.

() there iswithsuch thatuniformly a.e. in.

Then for any given, the problem (1.1) possesses at leastkpairs of nontrivial solutions.

In this section, we assume that and , is called a weak solution of the problem (1.1) if

Set

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 [9]) and such that

Now, we set , , so

Lemma 3.1Given, there issuch 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 existandsuch 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 subspaceWofand a constantsuch thatand.

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 ,

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 [9], 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).

1. Clément, PH, García-Huidobro, M, Manásevich, R, Schmitt, K: Mountain pass type solutions for quasilinear elliptic equations. Calc. Var. Partial Differ. Equ.. 11, 33–62 (2000). Publisher Full Text

2. Fukagai, N, Ito, M, Narukawa, MK: Positive solutions of quasilinear elliptic equations with critical Orlicz-Sobolev nonlinearity on . Funkc. Ekvacioj. 49, 235–267 (2006). Publisher Full Text

3. Fukagai, N, Narukawa, K: On the existence of multiple positive solutions of quasilinear elliptic eigenvalue problems. Ann. Mat. Pura Appl.. 186, 539–564 (2007). Publisher Full Text

4. García-Huidobro, M, Le, V, Manásevich, R, Schmitt, K: On the principal eigenvalues for quasilinear elliptic differential operators: an Orlicz-Sobolev space setting. Nonlinear Differ. Equ. Appl.. 6, 207–225 (1999). Publisher Full Text

5. Tan, Z, Fang, F: Orlicz-Sobolev versus Hölder local minimizer and multiplicity results for quasilinear elliptic equations. Preprint

6. Mihǎilescu, M, Rădulescu, V: Nonhomogeneous Neumann problems in Orlicz-Sobolev spaces. C. R. Math.. 346, 401–406 (2008). Publisher Full Text

7. Bonanno, G, Bisci, GM, Rǎdulescu, VD: Quasilinear elliptic non-homogeneous Dirichlet problems through Orlicz-Sobolev spaces. Nonlinear Anal.. 75, 4441–4456 (2012). Publisher Full Text

8. Černý, R: Generalized n-Laplacian: quasilinear nonhomogenous problem with critical growth. Nonlinear Anal.. 74, 3419–3439 (2011). Publisher Full Text

9. Fang, F, Tan, Z: Existence and multiplicity of solutions for a class of quasilinear elliptic equations: an Orlicz-Sobolev space setting. J. Math. Anal. Appl.. 389, 420–428 (2012). Publisher Full Text

10. Adams, RA, Fournier, JJF: Sobolev Spaces, Academic Press, Amsterdam (2003)

11. Ambrosetti, A, Rabinowitz, PH: Dual variational methods in critical point theory and applications. J. Funct. Anal.. 14, 349–381 (1973). Publisher Full Text

12. Bartolo, P, Benci, V, Fortunato, D: Abstract critical point theorems and applications to some nonlinear problems with “strong” resonance at infinity. Nonlinear Anal. TMA. 7, 981–1012 (1983). Publisher Full Text

13. Silva, EAB: Critical point theorems and applications to differential equations. PhD thesis, University of Wisconsin-Madison (1988)