Existence and multiplicity of solutions for equations involving nonhomogeneous operators of p(x)-Laplace type in RN

We are concerned with the following elliptic equations with variable exponents: −div(φ(x,∇u))+|u|p(x)−2u=λf(x,u) in RN, where the function φ(x,v) is of type |v|p(x)−2v with continuous function p:RN→(1,∞) and f:RN×R→R satisfies a Carathéodory condition. The purpose of this paper is to show the existence of at least one solution, and under suitable assumptions, infinitely many solutions for the problem above by using mountain pass theorem and fountain theorem.