Existence of solution for a class of quasilinear elliptic problem without 2 -condition

Abstract

The existence and multiplicity of solutions for a class of quasilinear elliptic problems are established for the type - φu = f(u)in ω,u = 0 on ω, where ω N, N ≥ 2, is a smooth bounded domain. The nonlinear term f: → is a continuous function which is superlinear at the origin and infinity. The function φ: → is an N-function where the well-known 2-condition is not assumed. Then the Orlicz-Sobolev space W01,φ(ω) may be non-reflexive. As a main model, we have the function φ(t) = (et2 - 1)/2,t ≥ 0. Here, we consider some situations where it is possible to work with global minimization, local minimization and mountain pass theorem. However, some estimates employed here are not standard for this type of problem taking into account the modular given by the N-function φ.