Existence of solution for a class of quasilinear problem in Orlicz-Sobolev space without ${\Delta }_2$-condition

Abstract

\noindent In this paper we study existence of solution for a class of problem of the type $$ \left\{ \begin{array}{ll} -Δ_Φ{u}=f(u), \quad \mbox{in} \quad Ωu=0, \quad \mbox{on} \quad \partial Ω, \end{array} \right. $$ where $\Omega \subset {\mathbb{R}}^N$, $N\ge 2$, is a smooth bounded domain, $f:\mathbb{R}\to \mathbb{R}$ is a continuous function verifying some conditions, and $\Phi :\mathbb{R}\to \mathbb{R}$ is a N-function which is not assumed to satisfy the well known ${\Delta }_2$-condition, then the Orlicz-Sobolev space $W_0^{1,\Phi }(\Omega )$ can be non reflexive. As main model we have the function $\Phi (t)=(e^{t^2}-1)/2$. Here, we study some situations where it is possible to work with global minimization, local minimization and mountain pass theorem, however some estimates are not standard for this type of problem.

arXiv commentIn this new version, we improve the Theorems 1.2 and 1.3, in the sense that we remove the assumption $2diam(Omega)\le 1$