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(phi( x, del u)) + vertical bar u vertical bar(p(x)?2)u = lambda f (x, u) in R-N, where the function phi(x, v) is of type vertical bar v vertical bar(p(x)?2)v with continuous function p : R-N -> (1,infinity) and f : R-N perpendicular to R -> R satis ? es a Carath ? odory condition. The purpose of this paper is to show the existence of at least one solution, and under suitable assumptions, in ? nitely many solutions for the problem above by using mountain pass theorem and fountain theorem.