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

Abstract

Abstract In this paper, we investigate the following nonlinear and non-homogeneous elliptic system: { − div ( a 1 ( | ∇ u | ) ∇ u ) + V 1 ( x ) a 1 ( | u | ) u = F u ( x , u , v ) in R N , − div ( a 2 ( | ∇ v | ) ∇ v ) + V 2 ( x ) a 2 ( | v | ) v = F v ( x , u , v ) in R N , ( u , v ) ∈ W 1 , Φ 1 ( R N ) × W 1 , Φ 2 ( R N ) , $$\begin{aligned} \textstyle\begin{cases} {-}\operatorname{div}(a_{1}( \vert \nabla{u} \vert )\nabla{u})+V_{1}(x)a_{1}( \vert u \vert )u=F_{u}(x,u,v)\quad \mbox{in } \mathbb{R}^{N},\\ {-}\operatorname{div}(a_{2}( \vert \nabla{v} \vert )\nabla{v})+V_{2}(x)a_{2}( \vert v \vert )v=F_{v}(x,u,v) \quad\mbox{in } \mathbb{R}^{N},\\ (u, v)\in W^{1,\Phi_{1}}(\mathbb{R}^{N})\times W^{1, \Phi_{2}}(\mathbb{R}^{N}), \end{cases}\displaystyle \end{aligned}$$ where ϕ i ( t ) = a i ( | t | ) t ( i = 1 , 2 ) $\phi_{i}(t)=a_{i}( \vert t \vert )t (i=1,2)$ are two increasing homeomorphisms from R $\mathbb{R}$ onto R $\mathbb{R}$ , functions V i ( i = 1 , 2 ) $V_{i}(i=1,2)$ and F are 1-periodic in x, and F satisfies some ( ϕ 1 , ϕ 2 ) $(\phi_{1},\phi_{2})$ -superlinear Orlicz-Sobolev conditions. By using a variant mountain pass lemma, we obtain that the system has a ground state.

Keywords

• Orlicz-Sobolev spaces
• quasilinear
• critical point
• ground state