On a class of logarithmic Schr\"odinger equations via perturbation method

TL;DR

This paper develops a novel perturbative variational method to prove the existence and multiplicity of solutions for a class of logarithmic Schrödinger equations with potential functions tending to infinity.

Contribution

It introduces a new perturbation approach to handle the non-smooth variational functional associated with the logarithmic Schrödinger equation.

Findings

Established existence of nontrivial weak solutions.

Proved multiplicity of solutions under specified conditions.

Overcame the challenge of non-^{1}-smoothness of the functional.

Abstract

In this paper, we consider the following logarithmic Schr\"odinger equation \[ -\Delta u + V(x)u = u \log u^{2},\quad x\in\mathbb{R}^{N}. \] Assuming that \(V\in C(\mathbb{R}^{N},\mathbb R)\), \(V\) is bounded away from zero, and \(V(x)\to+\infty\) as \(|x|\to\infty\), we develop a new perturbative variational approach to overcome the lack of \(C^{1}\)-smoothness of the associated functional and prove the existence and multiplicity of nontrivial weak solutions.