MULTIPLE SOLUTIONS FOR QUASILINEAR ELLIPTIC NEUMANN PROBLEMS IN ORLICZ-SOBOLEV SPACES NIKOLAOS HALIDIAS AND VY K. LE Received 15 October 2004 and in revised form 21 January 2005 We investigate the existence of multiple solutions to quasilinear elliptic problems containing Laplace like operators (φ-Laplacians). We are interested in Neumann boundary value problems and our main tool is Br´ zis-Nirenberg’s local linking theorem. e 1. Introduction In this paper, we consider the following elliptic problem with Neumann boundary condition, −div α ∇u(x) ∇u(x) = g(x,u) . on Ω () ∂u = 0 . on ∂Ω. ∂ν Here, Ω is a bounded domain with sufficiently smooth (. Lipschitz) boundary. | MULTIPLE SOLUTIONS FOR QUASILINEAR ELLIPTIC NEUMANN PROBLEMS IN ORLICZ-SOBOLEV spaces NIKOLAOS HALIDIAS AND VY K. LE Received 15 October 2004 and in revised form 21 January 2005 We investigate the existence of multiple solutions to quasilinear elliptic problems containing Laplace like operators ộ-Laplacians . We are interested in Neumann boundary value problems and our main tool is Brezis-Nirenberg s local linking theorem. 1. Introduction In this paper we consider the following elliptic problem with Neumann boundary condition -div a I Vu x I Vu x g x u . on D du 0 . on dD. dv Here D is a bounded domain with sufficiently smooth . Lipschitz boundary dD and d dv denotes the outward normal derivative on dD. We assume that the function ộ R R defined by ộ s a s s if s 0 and 0 otherwise is an increasing homeomorphism from R to R. Let O s Ị0ộ f dt s G R. Then is a Young function. We denote by L the Orlicz space associated with and by II II the usual Luxemburg norm on L u inf u x k 0 dx 1 . D k Also w 1L is the corresponding Orlicz-Sobolev space with the norm II u II o II u n Vu . The boundary value problem has the following weak formulation in w 1L u G w 1L J a Vu Vu Vvdx J g - u vdx Wv G w 1L . Our goal in this short note is to prove the existence of two nontrivial solutions to our problem under some suitable conditions on g. The main tool that we are going to use is an abstract existence result of Brezis and Nirenberg 1 which is stated here for the sake of completeness. Copyright 2006 Hindawi Publishing Corporation Boundary Value Problems 2005 3 2005 299-306 DOI 300 Multiple solutions for Neumann problems First let us recall the well known Palais-Smale PS condition. Let X be a Banach space and I X R. We say that I satisfies the PS condition if any sequence un Q X satisfying II un M 1 1 u 0 1 e 0 X n n n with en 0 has a convergent subsequence. Theorem 1 . Let X be a Banach space with a direct sum decomposition X X1 X2
