Multiple normalized solutions to a logarithmic Schrödinger equation via Lusternik-Schnirelmann category

Abstract

In this paper our objective is to investigate the existence of multiple normalized solutions to the logarithmic Schrödinger equation given by \begin{align*} \left\{ \begin{aligned} &-ε^2 Δu+V( x)u=λu+u \log u^2, \quad \quad \hbox{in }\mathbb{R}^N,\\ &\int_{\mathbb{R}^{N}}|u|^{2}dx=a^{2}ε^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 $V:{\mathbb{R}}^N\to [-1,\infty )$ 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 $C^1$.