Hostname: page-component-6766d58669-tq7bh Total loading time: 0 Render date: 2026-05-14T13:21:55.679Z Has data issue: false hasContentIssue false
  • English
  • Français

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

Part of: Existence theories Equations of mathematical physics and other areas of application Miscellaneous topics - Partial differential equations 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, 20133 Milano 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, 20133 Milano 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, 20133 Milano Italy (dario.pierotti@polimi.it, nicola.soave@polimi.it, gianmaria.verzini@polimi.it)
Get access Rights & Permissions [Opens in a new window]

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

Information

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 151 , Issue 2 , April 2021 , pp. 705 - 733
Check for updates
Copyright
Copyright © The Author(s), 2020. Published by Cambridge University Press on behalf of The Royal Society of Edinburgh

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

References

1Adami, R., Serra, E. and Tilli, P.. NLS ground states on graphs. Calc. Var. Partial Differ. Equ. 54 (2015), 743–761.CrossRefGoogle Scholar
2Adami, R., Serra, E. and Tilli, P.. Threshold phenomena and existence results for NLS ground states on metric graphs. J. Funct. Anal. 271 (2016), 201–223.CrossRefGoogle Scholar
3Adami, R., Serra, E. and Tilli, P.. Negative energy ground states for the L 2-critical NLSE on metric graphs. Comm. Math. Phys. 352 (2017), 387–406.CrossRefGoogle Scholar
4Adami, R., Serra, E. and Tilli, P.. Nonlinear dynamics on branched structures and networks. Riv. Math. Univ. Parma (N.S.) 8 (2017), 109–159.Google Scholar
5Adami, R., Serra, E. and Tilli, P.. Multiple positive bound states for the subcritical NLS equation on metric graphs. Calc. Var. Partial Differ. Equ. 58 (2019), 5.CrossRefGoogle Scholar
6Bellazzini, J., Boussaïd, N., Jeanjean, L. and Visciglia, N.. Existence and stability of standing waves for supercritical NLS with a partial confinement. Comm. Math. Phys. 353 (2017), 229–251.CrossRefGoogle Scholar
7Bellazzini, J., Georgiev, V. and Visciglia, N.. Long time dynamics for semi-relativistic NLS and half wave in arbitrary dimension. Math. Ann. 371 (2018), 707–740.CrossRefGoogle Scholar
8Bellazzini, J. and Jeanjean, L.. On dipolar quantum gases in the unstable regime. SIAM J. Math. Anal. 48 (2016), 2028–2058.CrossRefGoogle Scholar
9Berestycki, H. and Lions, P.-L.. Nonlinear scalar field equations. I. Existence of a ground state. Arch. Rational Mech. Anal. 82 (1983), 313–345.CrossRefGoogle Scholar
10Berestycki, H. and Lions, P.-L.. Nonlinear scalar field equations. II. Existence of infinitely many solutions. Arch. Rational Mech. Anal. 82 (1983), 347–375.CrossRefGoogle Scholar
11Berkolaiko, G. and Kuchment, P. 2013 Introduction to quantum graphs, Vol. 186, Mathematical Surveys and Monographs. RI, Providence: American Mathematical Society.Google Scholar
12Cazenave, T. 2003 Semilinear Schrödinger equations, Vol.10, Courant Lecture Notes in Mathematics. New York: New York University Courant Institute of Mathematical Sciences.Google Scholar
13Dovetta, S.. Existence of infinitely many stationary solutions of the L 2-subcritical and critical NLSE on compact metric graphs. J. Differ. Equ. 264 (2018), 4806–4821.CrossRefGoogle Scholar
14Dovetta, S.. Mass-constrained ground states of the stationary NLSE on periodic metric graphs. NoDEA Nonlinear Differ. Equ. Appl. 26 (2019), 30.CrossRefGoogle Scholar
15Dovetta, S. and Tentarelli, L.. L 2-critical NLS on noncompact metric graphs with localized nonlinearity: topological and metric features. Calc. Var. Partial Differ. Equ. 58 (2019), 108.CrossRefGoogle Scholar
16Grillakis, M., Shatah, J. and Strauss, W.. Stability theory of solitary waves in the presence of symmetry, I. J. Funct. Anal. 74 (1987), 160–197.CrossRefGoogle Scholar
17Grillakis, M., Shatah, J. and Strauss, W.. Stability theory of solitary waves in the presence of symmetry. II. J. Funct. Anal. 94 (1990), 308–348.CrossRefGoogle Scholar
18Noja, D.. Nonlinear Schrödinger equation on graphs: recent results and open problems. Philos. Trans. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 372 (2014), 20130002.Google ScholarPubMed
19Noja, D. and Pelinovsky, D. E.. Standing waves of the quintic NLS equation on the tadpole graph. Preprint arXiv:2001.00881, 2020.Google Scholar
20Noris, B., Tavares, H. and Verzini, G.. Existence and orbital stability of the ground states with prescribed mass for the L 2-critical and supercritical NLS on bounded domains. Anal. PDE 7 (2014), 1807–1838.CrossRefGoogle Scholar
21Noris, B., Tavares, H. and Verzini, G.. Stable solitary waves with prescribed L 2-mass for the cubic Schrödinger system with trapping potentials. Discrete Contin. Dyn. Syst. 35 (2015), 6085–6112.CrossRefGoogle Scholar
22Noris, B., Tavares, H. and Verzini, G.. Normalized solutions for nonlinear Schrödinger systems on bounded domains. Nonlinearity 32 (2019), 1044–1072.CrossRefGoogle Scholar
23Pierotti, D. and Verzini, G.. Normalized bound states for the nonlinear Schrödinger equation in bounded domains. Calc. Var. Partial Differ. Equ. 56 (2017), 133.CrossRefGoogle Scholar
24Serra, E. and Tentarelli, L.. Bound states of the NLS equation on metric graphs with localized nonlinearities. J. Differ. Equ. 260 (2016), 5627–5644.CrossRefGoogle Scholar
25Soave, N.. Normalized ground states for the NLS equation with combined nonlinearities. Preprint arXiv:1811.00826, 2018.Google Scholar
26Soave, N.. Normalized ground states for the NLS equation with combined nonlinearities: the Sobolev critical case. Preprint arXiv:1901.02003, 2019.Google Scholar