Quasilinear elliptic problems on non-reflexive Orlicz-Sobolev spaces

Abstract

In the paper the existence, uniqueness and the multiplicity of solutions for a quasilinear elliptic problems driven by the $\Phi$-Laplacian operator is established. Here we consider the non-reflexive case taking into account the Orlicz and Orlicz-Sobolev framework. The non-reflexive case occurs when the $N$-function $\widetilde{\Phi}$ does not verify the $\Delta_{2}$-condition. In order to prove our main results we employ variational methods, regularity results and truncation arguments.