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- Title
Multiple Normalized Solutions to a Logarithmic Schrödinger Equation via Lusternik–Schnirelmann Category.
- Authors
Alves, Claudianor O.; Ji, Chao
- Abstract
In this paper we will investigate the existence of multiple normalized solutions to the logarithmic Schrödinger equation given by - ϵ 2 Δ u + V (x) u = λ u + u log u 2 , in R N , ∫ R N | u | 2 d x = a 2 ϵ N , where N ≥ 1 , 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 C 1 .
- Subjects
SCHRODINGER equation; LAGRANGE multiplier; FUNCTION spaces; CONTINUOUS functions; TOPOLOGY
- Publication
Journal of Geometric Analysis, 2024, Vol 34, Issue 7, p1
- ISSN
1050-6926
- Publication type
Article
- DOI
10.1007/s12220-024-01649-y