Multiple normalized solutions to a logarithmic Schr\"{o}dinger equation via Lusternik-Schnirelmann category

TL;DR

This paper proves the existence of multiple normalized solutions to a logarithmic Schrödinger equation by linking the number of solutions to the topology of the potential's minimum set, using Lusternik-Schnirelmann theory.

Contribution

It introduces a new function space for the energy functional and establishes multiple solutions based on topological methods, which is a novel approach for this equation.

Findings

Number of solutions related to the topology of the potential's minimum set

Existence of multiple solutions proved using Lusternik-Schnirelmann category

A new function space where the energy functional is of class C^1 introduced

Abstract

In this paper our objective is to investigate the existence of multiple normalized solutions to the logarithmic Schr\"{o}dinger equation given by \begin{align*} \left\{ \begin{aligned} &-\epsilon^2 \Delta u+V( x)u=\lambda u+u \log u^2, \quad \quad \hbox{in }\mathbb{R}^N,\\ &\int_{\mathbb{R}^{N}}|u|^{2}dx=a^{2}\epsilon^N, \end{aligned} \right. \end{align*} where $a, \epsilon>0, \lambda \in \mathbb{R}$ is an unknown parameter that appears as a Lagrange multiplier and $V: \mathbb{R}^N \rightarrow[-1, \infty)$ is a continuous function. Our analysis demonstrates that the number of normalized solutions of the equation is associated with the topology of the set where the potential function $V$ attains its minimum value. To prove the main result, we employ minimization techniques and use the Lusternik-Schnirelmann category. Additionally, we introduce a new function space where the energy functional associated with the problem is of class $C^1$.