Existence of solution for two classes of quasilinear systems defined on a non-reflexive Orlicz-Sobolev Spaces

This paper proves the existence of nontrivial solution for two classes of quasilinear systems of the type \begin{equation*} \left{; \begin{aligned} -Δ_{Φ_{1}} u&=F_u(x,u,v)+λR_u(x,u,v);$\text{ in }$ Ω& \ -Δ_{Φ_{2}} v&=-F_v(x,u,v)-λR_v(x,u,v);$\text{ in }$ Ω& \ u=v&=0;$\text{ on }$ \partialΩ& \end{aligned} \right. \end{equation*} where $\lambda >0$ is a parameter, $\mathrm{\Omega }$ is a bounded domain in ${\mathbb{R}}^N$($N\ge 2$) with smooth boundary $\mathrm{\partial }\mathrm{\Omega }$. The first class we drop the ${\mathrm{\Delta }}_2$-condition of the functions ${\tilde{\mathrm{\Phi }}}_i$($i=1,2$) and assume that $F$ has a double criticality. For this class, we use a linking theorem without the Palais-Smale condition for locally Lipschitz functionals combined with a concentration-compactness lemma for nonreflexive Orlicz-Sobolev space. The second class, we relax the ${\mathrm{\Delta }}_2$-condition of the functions ${\mathrm{\Phi }}_i$($i=1,2$). For this class, we consider $F=0$ and $\lambda =1$ and obtain the proof based on a saddle-point theorem of Rabinowitz without the Palais-Smale condition for functionals Frechet differentiable combined with some properties of the weak${}^{\ast }$ topology.