[Existence of multiple solutions for a Schr\"ondiger logarithmic equation

TL;DR

This paper proves the existence of multiple solutions for a Schrödinger logarithmic equation in  space, using Lusternik-Schnirelmann category and a new function space to analyze the energy functional.

Contribution

It introduces a new function space making the energy functional ^1 and applies topological methods to establish multiple solutions.

Findings

Multiple solutions exist for the logarithmic Schrd6dinger equation.

A new function space is constructed where the energy functional is ^1.

Lusternik-Schnellmann category is used to count solutions.

Abstract

This paper concerns the existence of multiple solutions for a Schr\"odinger logarithmic equation of the form \begin{equation} \left\{\begin{aligned} -\varepsilon^2\Delta u + V(x)u & =u\log u^2,\;\;\mbox{in}\;\;\mathbb{R}^{N},\nonumber u \in H^{1}(\mathbb{R}^{N}), \end{aligned} \right.\leqno{(P_\varepsilon)} \end{equation} where $V:\mathbb{R}^N\longrightarrow \mathbb{R}$ is a continuous function that satisfies some technical conditions and $\varepsilon$ is a positive parameter. We will establish the multiplicity of solution for $(P_\varepsilon)$ by using the notion of Lusternik-Schnirelmann category, by introducing a new function space where the energy functional is $C^1$.