Standing wave solutions of a perturbed Schrodinger equation in an Orlicz-Sobolev space setting

Abstract

In this paper we study a class of non-linear Schr\" odinger-type equation on the whole space. The differential operator was introduced by A. Azzollini {\em et al.} and it is described by a potential with a different growth near zero and at infinity.

We have the following perturbed problem:
$$\textrm{-div}\left[\phi'(|\nabla u|^2)\nabla u\right]+ \gamma(x)|u|^{\alpha-2}u=K(x)|u|^{s-2}u+g(x)\qquad \mbox{ in }\;\:\mathbb{R}^N$$

Our aim is to show that if the perturbation $g(x)$ is not too large in a suitable topology, we can prove the existence of at least one nontrivial solution using the mountain pass theorem in the framework of an Orlicz-Sobolev space.