Existence of solutions for a variable exponent system without PS conditions

Abstract

In this article, we study the existence of solution for the following elliptic system of variable exponents with perturbation terms $$\displaylines{ -\hbox{div}| \nabla u| ^{p(x)-2}\nabla u)+|u| ^{p(x)-2}u =\lambda a(x)| u| ^{\gamma(x)-2}u+F_{u}(x,u,v)\quad\hbox{in } \mathbb{R}^N, \\ -\hbox{div}| \nabla v| ^{q(x)-2}\nabla v)+|v| ^{q(x)-2}v =\lambda b(x)| v| ^{\delta(x)-2}v+F_{v}(x,u,v)\quad \hbox{in }\mathbb{R}^N, \\ u\in W^{1,p(\cdot )}(\mathbb{R}^N),v\in W^{1,q(\cdot )}(\mathbb{R}^N), }$$ where the corresponding functional does not satisfy PS conditions. We obtain a sufficient condition for the existence of solution and also present a result on asymptotic behavior of solutions at infinity.

Keywords

• Variable exponent system
• integral functional
• PS condition
• variable exponent Sobolev space