Existence of solution for a class of quasilinear problem in Orlicz-Sobolev space without $\Delta_2$-condition

\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.
Comment: In this new version, we improve the Theorems 1.2 and 1.3, in the sense that we remove the assumption $2diam(Omega)\leq 1$