Existence of ground state solutions for a class of quasilinear elliptic systems in Orlicz-Sobolev spaces

In this paper, we investigate the following nonlinear and non-homogeneous elliptic system: {−div(a1(|∇u|)∇u)+V1(x)a1(|u|)u=Fu(x,u,v)in RN,−div(a2(|∇v|)∇v)+V2(x)a2(|v|)v=Fv(x,u,v)in RN,(u,v)∈W1,Φ1(RN)×W1,Φ2(RN),\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\begin{aligned} \textstyle\begin{cases} {-}\operatorname{div}(a_{1}( \vert \nabla{u} \vert )\nabla{u})+V_{1}(x)a_{1}( \vert u \vert )u=F_{u}(x,u,v)\quad \mbox{in } \mathbb{R}^{N},\\ {-}\operatorname{div}(a_{2}( \vert \nabla{v} \vert )\nabla{v})+V_{2}(x)a_{2}( \vert v \vert )v=F_{v}(x,u,v) \quad\mbox{in } \mathbb{R}^{N},\\ (u, v)\in W^{1,\Phi_{1}}(\mathbb{R}^{N})\times W^{1, \Phi_{2}}(\mathbb{R}^{N}), \end{cases}\displaystyle \end{aligned}$$ \end{document} where ϕi(t)=ai(|t|)t(i=1,2)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\phi_{i}(t)=a_{i}( \vert t \vert )t (i=1,2)$\end{document} are two increasing homeomorphisms from R\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathbb{R}$\end{document} onto R\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathbb{R}$\end{document}, functions Vi(i=1,2)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$V_{i}(i=1,2)$\end{document} and F are 1-periodic in x, and F satisfies some (ϕ1,ϕ2)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$(\phi_{1},\phi_{2})$\end{document}-superlinear Orlicz-Sobolev conditions. By using a variant mountain pass lemma, we obtain that the system has a ground state.