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

TL;DR

This paper proves the existence of positive solutions for a class of quasilinear Schrödinger equations in nonreflexive Orlicz-Sobolev spaces, extending Hardy inequalities and using variational methods for non-differentiable functionals.

Contribution

It extends Hardy-type inequalities to nonreflexive Orlicz spaces and establishes ground state solutions for quasilinear Schrödinger equations with potential vanishing at infinity.

Findings

Existence of positive solutions in nonreflexive Orlicz-Sobolev spaces.

Extension of Hardy inequalities to nonreflexive Orlicz spaces.

Application of variational methods to non-differentiable functionals.

Abstract

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 $$- \Delta_{\Phi} u +V(x)\phi(|u|)u= K(x)f(u)\mbox{ in } \mathbb{R}^{N}$$ where $N\geq2$, $V,K$ are nonnegative continuous functions and $f$ is a continuous function with a quasicritical growth. Here we extend the Hardy-type inequalities presented in \cite{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 $V=0$.