Existence and orbital stability/instability of standing waves with prescribed mass for the L2-supercritical NLS in bounded domains and exterior domains

Relying on bifurcation type arguments, we obtain threshold results for the existence, non-existence and multiplicity of positive solutions with prescribed L 2 norm for the semi-linear elliptic equation in bounded domains: - Δ u - f ( x , u ) = λ u , x ∈ Ω , u | ∂ Ω = 0 , and we do not assume that f...

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Bibliographic Details
Published in:Calculus of variations and partial differential equations 2023, Vol.62 (6)
Main Author: Song, Linjie
Format: Article
Language:English
Subjects:
Analysis
Calculus of Variations and Optimal Control
Optimization
Control
Domains
Lower bounds
Mathematical and Computational Physics
Mathematics
Mathematics and Statistics
Orbital stability
Standing waves
Systems Theory
Theoretical
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Summary:Relying on bifurcation type arguments, we obtain threshold results for the existence, non-existence and multiplicity of positive solutions with prescribed L 2 norm for the semi-linear elliptic equation in bounded domains: - Δ u - f ( x , u ) = λ u , x ∈ Ω , u | ∂ Ω = 0 , and we do not assume that f is autonomous. Moreover, we provide a lower bound for the threshold. For almost every L 2 mass in the existence range, there exist one orbitally stable standing wave and one unstable standing wave associated to these positive solutions. We also study the existence of prescribed norm solutions in exterior domains and orbitally unstable standing waves for almost every L 2 mass in the existence range.
ISSN:0944-2669
1432-0835
DOI:10.1007/s00526-023-02510-w