Abstract
We study the nonlinear boundary value problem - div ((a(1)(vertical bar del u(x)vertical bar) + + a(2)(vertical bar del u (x)vertical bar)) del u (x)) = lambda vertical bar u vertical bar(q (x) -2) u - mu vertical bar u vertical bar(alpha(x)-2) u in Omega, u = 0 on partial derivative Omega, where Omega is a bounded domain in R-N with smooth boundary, lambda, mu are positive real numbers, q and alpha are continuous functions and a(1), a(2) are two mappings such that a(1) (vertical bar t vertical bar) t; a(2) (vertical bar t vertical bar) t; are increasing homeomorphisms from R to R. The problem is analysed in the context of Orlicz-Soboev spaces. First we show the existence of infinitely many weak solutions for any lambda; mu > 0. Second we prove that for any mu > 0, there exists lambda(*) sufficiently small, and lambda* large enough such that for any lambda is an element of ( 0, lambda(*)) boolean OR (lambda*, infinity), the above nonhomogeneous quasilinear problem has a non-trivial weak solution.