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

2023/07/03 by Claudianor O. Alves, Chao Ji, Alves, Claudianor O. +1
Computer Science · Mathematics · #Advanced Mathematical Modeling in Engineering #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #FOS: Mathematics #Nonlinear Partial Differential Equations

paper · pdf · doi:10.48550/arxiv.2307.01127

openalex publication_date 2023/07/03 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

In this paper our objective is to investigate the existence of multiple normalized solutions to the logarithmic Schrödinger equation given by \ \beginaligned amp;-ε² Δu+V( x)u=λu+u log u², \hboxin ℝ^N,
amp;∫_ℝ^N|u|²dx=a²ε^N, \endaligned . where a, ε>0, λ∈ ℝ is an unknown parameter that appears as a Lagrange multiplier and V: ℝ^N →[-1, ∞) 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 C¹.

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