Existence of positive solution for a class of quasilinear Schrödinger equations with potential vanishing at infinity on nonreflexive Orlicz-Sobolev spaces

In this paper we investigate the existence of positive solution for a class of quasilinear problem on an Orlicz-Sobolev space that can be nonreflexive - ΔΦ u +V(x)φ(|u|)u= K(x)f(u) RN, where N ≥ 2, $ V, K $ are nonnegative continuous functions and f is a continuous function with a quasicritical growth. Here we extend the Hardy-type inequalities presented in {AlvesandMarco} to nonreflexive Orlicz spaces. Through inequalities together with a variational method for non-differentiable functionals we will obtain a ground state solution. We analyze also the problem with $V=0$.