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:
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.,
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:
and is equipped with the (Luxemburg) norm, i.e.,
where we suppose that
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.
Using the version of the symmetric mountain pass theorem mentioned above, we can state our result as follows.
3 Main results and proofs
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
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. □
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 .
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).
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
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
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
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
Mihǎilescu, M, Rădulescu, V: Nonhomogeneous Neumann problems in Orlicz-Sobolev spaces. C. R. Math.. 346, 401–406 (2008). Publisher Full Text
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
Černý, R: Generalized n-Laplacian: quasilinear nonhomogenous problem with critical growth. Nonlinear Anal.. 74, 3419–3439 (2011). Publisher Full Text
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
Ambrosetti, A, Rabinowitz, PH: Dual variational methods in critical point theory and applications. J. Funct. Anal.. 14, 349–381 (1973). Publisher Full Text
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