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Local minimizers in absence of ground states for the critical NLS energy on metric graphs

Part of: Miscellaneous topics - Partial differential equations Existence theories Equations of mathematical physics and other areas of application General mathematical topics and methods in quantum theory

Published online by Cambridge University Press:  22 May 2020

Dario Pierotti ,
Nicola Soave  and
Gianmaria Verzini
Dario Pierotti
Affiliation:
Dipartimento di Matematica, Politecnico di Milano, piazza Leonardo da Vinci 32, 20133Milano Italy (dario.pierotti@polimi.it, nicola.soave@polimi.it, gianmaria.verzini@polimi.it)
Nicola Soave
Affiliation:
Dipartimento di Matematica, Politecnico di Milano, piazza Leonardo da Vinci 32, 20133Milano Italy (dario.pierotti@polimi.it, nicola.soave@polimi.it, gianmaria.verzini@polimi.it)
Gianmaria Verzini
Affiliation:
Dipartimento di Matematica, Politecnico di Milano, piazza Leonardo da Vinci 32, 20133Milano Italy (dario.pierotti@polimi.it, nicola.soave@polimi.it, gianmaria.verzini@polimi.it)
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Abstract

We consider the mass-critical non-linear Schrödinger equation on non-compact metric graphs. A quite complete description of the structure of the ground states, which correspond to global minimizers of the energy functional under a mass constraint, is provided by Adami, Serra and Tilli in [R. Adami, E. Serra and P. Tilli. Negative energy ground states for the L2-critical NLSE on metric graphs. Comm. Math. Phys. 352 (2017), 387–406.] , where it is proved that existence and properties of ground states depend in a crucial way on both the value of the mass, and the topological properties of the underlying graph. In this paper we address cases when ground states do not exist and show that, under suitable assumptions, constrained local minimizers of the energy do exist. This result paves the way to the existence of stable solutions in the time-dependent equation in cases where the ground state energy level is not achieved.

Keywords

Normalized solutionsnon-compact metric graphsnon-linear Schrödinger equationL2-critical exponent

MSC classification

Primary: 35R02: Partial differential equations on graphs and networks (ramified or polygonal spaces)
Secondary: 35Q55: NLS-like equations (nonlinear Schrödinger) 81Q35: Quantum mechanics on special spaces 49J40: Variational methods including variational inequalities
Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 151 , Issue 2 , April 2021 , pp. 705 - 733
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Copyright
Copyright © The Author(s), 2020. Published by Cambridge University Press on behalf of The Royal Society of Edinburgh

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