Existence of solution for a class of quasilinear elliptic problem without Δ2-condition

Abstract

The existence and multiplicity of solutions for a class of quasilinear elliptic problems are established for the type [Formula: see text] where [Formula: see text], [Formula: see text], is a smooth bounded domain. The nonlinear term [Formula: see text] is a continuous function which is superlinear at the origin and infinity. The function [Formula: see text] is an [Formula: see text]-function where the well-known [Formula: see text]-condition is not assumed. Then the Orlicz–Sobolev space [Formula: see text] may be non-reflexive. As a main model, we have the function [Formula: see text]. 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 [Formula: see text]-function [Formula: see text].