\noindent In this paper we study existence of solution for a class of problem of the type $$ \left\{ \begin{ array}{ ll} -\Delta_{ \Phi}{ u}=f(u), \quad \mbox{ in} \quad \Omega u=0, \quad \mbox{ on} \quad \partial \Omega, \end{ array} \right. $$ where $\Omega \subset \mathbb{ R}^N$, $N \geq 2$, 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 $W^{ 1,\Phi}_0(\Omega)$ 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.
Categories: math.AP