A class of elliptic system in non reflexive Orlicz-Sobolev spaces

This paper aims to show that there exists a weak solution to the following quasilinear system driven by the M-Laplacian 0.1 \(\begin{aligned} {\left\{ \begin{array}{ll} (-\Delta _{m_1})u=F_u(x,u,v)& in\quad \Omega , \\ (-\Delta _{m_2})v=F_v(x,u,v)& in\quad \Omega ,\\ u=v=0& in\quad \partial \Omega , \end{array}\right. } \end{aligned}\) $$\begin{array}{c}\hfill \left\{\begin{array}{cc}(-{\mathrm{\Delta }}_{m_1})u=F_u(x,u,v)\hfill & in\mathrm{\Omega },\hfill \\ (-{\mathrm{\Delta }}_{m_2})v=F_v(x,u,v)\hfill & in\mathrm{\Omega },\hfill \\ u=v=0\hfill & in\partial \mathrm{\Omega },\hfill \end{array}\right)\end{array}$$ where \(\Omega \) $\mathrm{\Omega }$ is a bounded open subset in \({\mathbb {R}}^N\) ${\mathbb{R}}^N$ and \((-\Delta _{m})\) $(-{\mathrm{\Delta }}_m)$ is the M-Laplacian operator. Here we consider the non-reflexive case taking into account the Orlicz and Orlicz-Sobolev Space. The non-reflexive case occurs when the N-function \({\overline{M}}\) $\overline{M}$ does not verify the \(\Delta _2\) ${\mathrm{\Delta }}_2$ -condition. We consider an approximated quasilinear elliptic problem driven by the \(M_\epsilon \) $M_{ϵ}$ -Laplacian and using the Mountain Pass Theorem to obtain the existence of a nontrivial and nonnegative solution for the above system in reflexive case. By tending \(\epsilon \rightarrow 0\) $ϵ\to 0$ we get the solution in the non-reflexive case.