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

Abstract

In this paper, we investigate the following nonlinear and non-homogeneous elliptic system: \t\t\t{−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}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\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 phi_{i}(t)=a_{i}( vert t vert )t (i=1,2) are two increasing homeomorphisms from mathbb{R} onto mathbb{R}, functions V_{i}(i=1,2) and F are 1-periodic in x, and F satisfies some (phi_{1},phi_{2})-superlinear Orlicz-Sobolev conditions. By using a variant mountain pass lemma, we obtain that the system has a ground state.