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

Abstract

This paper proves the existence of nontrivial solution for two classes of quasilinear systems of the type \begin{equation*} \left\{\; \begin{aligned} -\Delta_{\Phi_{1}} u&=F_u(x,u,v)+\lambda R_u(x,u,v)\;\text{ in } \Omega& \\ -\Delta_{\Phi_{2}} v&=-F_v(x,u,v)-\lambda R_v(x,u,v)\;\text{ in } \Omega& \\ u=v&=0\;\text{ on } \partial\Omega& \end{aligned} \right. \end{equation*} where $\lambda > 0$ is a parameter, $\Omega$ is a bounded domain in $\mathbb{R}^N$($N \geq 2$) with smooth boundary $\partial \Omega$. The first class we drop the $\Delta_2$-condition of the functions $\tilde{\Phi}_i$($i=1,2$) 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 $\Delta_2$-condition of the functions ${\Phi}_i$($i=1,2$). 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$^*$ topology.