Normalized solutions for a coupled Schrödinger system

By Thomas Bartsch, X. Zhong, Wenming Zou. Mathematische Annalen, 2019

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In the present paper, we prove the existence of solutions (λ1,λ2,u,v)∈R2×H1(R3,R2)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(\lambda _1,\lambda _2,u,v)\in \mathbb {R}^2\times H^1(\mathbb {R}^3,\mathbb {R}^2)$$\end{document} to systems of coupled Schrödinger equations -Δu+λ1u=μ1u3+βuv2inR3-Δv+λ2v=μ2v3+βu2vinR3u,v>0inR3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} {\left\{ \begin{array}{ll} -\Delta u+\lambda _1u=\mu _1 u^3+\beta uv^2\quad &{}\hbox {in}\;\mathbb {R}^3\\ -\Delta v+\lambda _2v=\mu _2 v^3+\beta u^2v\quad &{}\hbox {in}\;\mathbb {R}^3\\ u,v>0&{}\hbox {in}\;\mathbb {R}^3 \end{array}\right. } \end{aligned}$$\end{document}satisfying the normalization constraint ∫R3u2=a2and∫R3v2=b2,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \int _{\mathbb {R}^3}u^2=a^2\quad \hbox {and}\;\int _{\mathbb {R}^3}v^2=b^2, $$\end{document} which appear in binary mixtures of Bose–Einstein condensates or in nonlinear optics. The parameters μ1,μ2,β>0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mu _1,\mu _2,\beta >0$$\end{document} are prescribed as are the masses a,b>0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$a,b>0$$\end{document}. The system has been considered mostly in the case of fixed frequencies λ1,λ2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lambda _1,\lambda _2$$\end{document}. When the masses are prescribed, the standard approach to this problem is variational with λ1,λ2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lambda _1,\lambda _2$$\end{document} appearing as Lagrange multipliers. Here we present a new approach based on the fixed point index in cones, bifurcation theory, and the continuation method. We obtain the existence of normalized solutions for any given a,b>0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$a,b>0$$\end{document} for β\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\beta $$\end{document} in a large range. We also have a result about the nonexistence of positive solutions which shows that our existence theorem is almost optimal. Especially, if μ1=μ2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mu _1=\mu _2$$\end{document} we prove that normalized solutions exist for all β>0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\beta >0$$\end{document} and all a,b>0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$a,b>0$$\end{document}.

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

View Paper: https://www.semanticscholar.org/paper/9e10186cf58741eeda00164f87bfca38aa749753

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Citations in the corpus

Normalized solutions to mass supercritical Schrödinger equations with negative potential (2021, methodology): “To conclude this discussion about the previous literature, since the results we have discussed so far are all of variational nature, let us mention that also topological methods have been applied, for instance in [15, 10, 31], also in connection with ergodic Mean Field Games systems.”
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.”
A global branch approach to normalized solutions for the Schrödinger equation (2021, background): “The results of [6] significantly extended the ones of Bartsch et al.”