A logarithmic Schr\"{o}dinger equation with asymptotic conditions on the potential

TL;DR

This paper studies a class of logarithmic Schrödinger equations with sign-changing potentials, establishing existence of infinitely many solutions for coercive potentials and a ground state for bounded potentials using variational methods.

Contribution

It introduces new existence results for solutions of logarithmic Schrödinger equations with sign-changing potentials, employing the Fountain theorem and variational techniques.

Findings

Infinitely many solutions for coercive potentials.

Existence of a ground state for bounded potentials.

Development of variational methods for non-smooth functionals.

Abstract

In this paper we consider a class of logarithmic Schr\"{o}dinger equations with a potential which may change sign. When the potential is coercive, we obtain infinitely many solutions by adapting some arguments of the Fountain theorem, and in the case of bounded potential we obtain a ground state solution, i.e. a nontrivial solution with least possible energy. The functional corresponding to the problem is the sum of a smooth and a convex lower semicontinuous term.