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Instability of ground states for the NLS equation with potential on the star graph

Instability of ground states for the NLS equation with potential on the star graph We study the nonlinear Schrödinger equation with an arbitrary real potential V(x)∈(L1+L∞)(Γ)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$V(x)\in (L^1+L^\infty )(\Gamma )$$\end{document} on a star graph Γ\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\Gamma $$\end{document}. At the vertex an interaction occurs described by the generalized Kirchhoff condition with strength -γ<0\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$-\gamma <0$$\end{document}. We show the existence of ground states φω(x)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\varphi _{\omega }(x)$$\end{document} as minimizers of the action functional on the Nehari manifold under additional negativity and decay conditions on V(x). Moreover, for V(x)=-βxα\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$V(x)=-\dfrac{\beta }{x^{\alpha }}$$\end{document}, in the supercritical case, we prove that the standing waves eiωtφω(x)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$e^{i\omega t}\varphi _{\omega }(x)$$\end{document} are orbitally unstable in H1(Γ)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$H^{1}(\Gamma )$$\end{document} when ω\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\omega $$\end{document} is large enough. Analogous result holds for an arbitrary γ∈R\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\gamma \in {\mathbb {R}}$$\end{document} when the standing waves have symmetric profile. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Journal of Evolution Equations Springer Journals

Instability of ground states for the NLS equation with potential on the star graph

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Publisher
Springer Journals
Copyright
Copyright © The Author(s), under exclusive licence to Springer Nature Switzerland AG part of Springer Nature 2021. corrected publication 2021
ISSN
1424-3199
eISSN
1424-3202
DOI
10.1007/s00028-021-00670-w
Publisher site
See Article on Publisher Site

Abstract

We study the nonlinear Schrödinger equation with an arbitrary real potential V(x)∈(L1+L∞)(Γ)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$V(x)\in (L^1+L^\infty )(\Gamma )$$\end{document} on a star graph Γ\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\Gamma $$\end{document}. At the vertex an interaction occurs described by the generalized Kirchhoff condition with strength -γ<0\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$-\gamma <0$$\end{document}. We show the existence of ground states φω(x)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\varphi _{\omega }(x)$$\end{document} as minimizers of the action functional on the Nehari manifold under additional negativity and decay conditions on V(x). Moreover, for V(x)=-βxα\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$V(x)=-\dfrac{\beta }{x^{\alpha }}$$\end{document}, in the supercritical case, we prove that the standing waves eiωtφω(x)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$e^{i\omega t}\varphi _{\omega }(x)$$\end{document} are orbitally unstable in H1(Γ)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$H^{1}(\Gamma )$$\end{document} when ω\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\omega $$\end{document} is large enough. Analogous result holds for an arbitrary γ∈R\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\gamma \in {\mathbb {R}}$$\end{document} when the standing waves have symmetric profile.

Journal

Journal of Evolution EquationsSpringer Journals

Published: Dec 1, 2021

Keywords: Nonlinear Schrödinger equation; Linear potential; Generalized Kirchhoff’s condition; Ground state; Orbital stability; Primary 35Q55; Secondary 35Q40

References