Multiple normalized solutions for a competing system of Schrödinger equations

By Thomas Bartsch, Nicola Soave. Calculus of Variations and Partial Differential Equations, 2017

239 citations recorded by the source corpus.

We prove the existence of infinitely many solutions $$\lambda _1, \lambda _2 \in \mathbb {R}$$λ1,λ2∈R, $$u,v \in H^1(\mathbb {R}^3)$$u,v∈H1(R3), for the nonlinear Schrödinger system $$\begin{aligned} {\left\{ \begin{array}{ll} -\Delta u - \lambda _1 u = \mu u^3+ \beta u v^2 &{} \text {in }\mathbb {R}^3 \\ -\Delta v- \lambda _2 v = \mu v^3 +\beta u^2 v &{} \text {in }\mathbb {R}^3\\ u,v>0 &{} \text {in }\mathbb {R}^3 \\ \int _{\mathbb {R}^3} u^2 = a^2 \quad \text {and} \quad \int _{\mathbb {R}^3} v^2 = a^2, \end{array}\right. } \end{aligned}$$-Δu-λ1u=μu3+βuv2inR3-Δv-λ2v=μv3+βu2vinR3u,v>0inR3∫R3u2=a2and∫R3v2=a2,where $$a,\mu >0$$a,μ>0 and $$\beta \le -\mu $$β≤-μ are prescribed. Our solutions satisfy $$u\ne v$$u≠v so they do not come from a scalar equation. The proof is based on a new minimax argument, suited to deal with normalization conditions.

Subfields: Applied Mathematics, Control and Systems Engineering, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/4da9ce3acecf3eedf9ba7e736536725c902d181a

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Citations in the corpus

Existence and asymptotic behavior of normalized ground states for Sobolev critical Schrödinger systems (2022, methodology): “We refer the reader to [7, 26, 27, 28, 29, 30, 39, 40, 43] for scalar equations, [8, 9, 10, 11, 14, 21, 22, 31] for systems of two equations, [35] for systems of k equations.”
Normalized solutions for a Schrödinger equation with critical growth in RN\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{documen (2021, unknown): “For the reader interested in normalized solutions for the Schrödinger equations, we would also like to refer [9,11,12,17,25,29,31,34,38,39,41] and references therein.”
Multiple positive normalized solutions for nonlinear Schrödinger systems (2017, background): “Note however that, when β < 0 the existence of infinitely many positive solutions was established in [7].”
Normalized solutions to mass supercritical Schrödinger equations with negative potential (2021, background): “After [22], only recently more general autonomous equations and systems have been considered, also refining and developing this initial strategy, see [4, 6, 8, 5, 17, 9, 21, 34, 35] and references therein.”