Existence and multiplicity of solutions for a class of (ϕ1,ϕ2ϕ1,ϕ2)-Laplacian elliptic system in RNRN via genus theory

In this paper, we investigate the following nonlinear and non-homogeneous elliptic system involving (ϕ1,ϕ2)(ϕ1,ϕ2)-Laplacian {−div(ϕ1(|∇u|)∇u)+V1(x)ϕ1(|u|)u=Fu(x,u,v)in  RN,−div(ϕ2(|∇v|)∇v)+V2(x)ϕ2(|v|)v=Fv(x,u,v)in  RN,(u,v)∈W1,Φ1(RN)×W1,Φ2(RN)with  N≥2, where the functions Vi(x)(i=1,2)Vi(x)(i=1,2) are bounded and positive in RNRN, the functions ϕi(t)t(i=1,2)ϕi(t)t(i=1,2) are increasing homeomorphisms from R+R+ onto R+R+, and the function FF is of class C1(RN+2,R)C1(RN+2,R) and has a sub-linear Orlicz–Sobolev growth. By using the least action principle, we obtain that system has at least one nontrivial solution. When FF satisfies an additional symmetric condition, by using the genus theory, we obtain that system has infinitely many solutions.