Multiple Normalized Solutions to a Logarithmic Schrödinger Equation via Lusternik–Schnirelmann Category

Abstract

In this paper we will investigate the existence of multiple normalized solutions to the logarithmic Schrödinger equation given by

$$\begin{aligned} \left\{ \begin{aligned}&-\epsilon ^2 \Delta u+V( x)u=\lambda u+u \log u^2, \quad \quad \hbox {in }\mathbb {R}^N,\\&\int _{\mathbb {R}^{N}}|u|^{2}dx=a^{2}\epsilon ^N, \end{aligned} \right. \end{aligned}$$

where $N\ge 1$, $a, \epsilon >0, \lambda \in \mathbb {R}$ 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 $C^1$.