A natural constraint approach to normalized solutions of nonlinear Schrödinger equations and systems

2017/01/01 by Thomas Bartsch, Nicola Soave · 30 citations
Mathematics · Physics and Astronomy · #Advanced Mathematical Physics Problems #Applied mathematics #Combinatorics #Constraint (computer-aided design) #Geometry #Lambda #Mathematical analysis #Mathematical physics #Mathematical proof #Mathematics #Nonlinear Waves and Solitons #Nonlinear system #Normalization (sociology) #Physics #Quantum mechanics #Scalar (mathematics) #Spectral Theory in Mathematical Physics

paper · open access · doi:10.1016/j.jfa.2017.01.025

published in Journal of Functional Analysis 272(12), 4998-5037 (Elsevier BV)

openalex publication_date 2017/02/07 · openalex created_date 2022/09/26 · openalex updated_date 2026/08/05

Abstract

The paper deals with the existence of normalized solutions to the system −Δu−λ1u=Î1⁄41u3+Î2uv2in R3−Δv−λ2v=Î1⁄42v3+Î2u2vin R3∫R3u2=a12and∫R3v2=a22 for any Î1⁄41,Î1⁄42,a1,a2>0 and Î2<0 prescribed. We present a new approach that is based on the introduction of a natural constraint associated to the problem. We also show that, as Î2→−∞, phase separation occurs for the solutions that we find. Our method can be adapted to scalar nonlinear Schrödinger equations with normalization constraint, and leads to alternative and simplified proofs to some results already available in the literature.

Citations

• Uniform Bounds for Strongly Competing Systems: The Optimal Lipschitz Case
• Uniform Hölder Bounds for Nonlinear Schrödinger Systems with Strong Competition

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