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Existence, non-existence and blow-up behaviour of minimizers for the mass-critical fractional non-linear Schrödinger equations with periodic potentials

Part of: Partial differential equations Equations of mathematical physics and other areas of application Elliptic equations and systems

Published online by Cambridge University Press:  13 January 2020

Van Duong Dinh
Van Duong Dinh*
Affiliation:
Laboratoire Paul Painlevé UMR 8524, Université de Lille CNRS, 59655 Villeneuve d'Ascq Cedex, France Department of Mathematics, HCMC University of Pedagogy, 280 An Duong Vuong, Ho Chi Minh, Vietnam (contact@duongdinh.com)
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Abstract

We consider the minimizing problem for the energy functional with prescribed mass constraint related to the fractional non-linear Schrödinger equation with periodic potentials. Using the concentration-compactness principle, we show a complete classification for the existence and non-existence of minimizers for the problem. In the mass-critical case, under a suitable assumption of the potential, we give a detailed description of blow-up behaviour of minimizers once the mass tends to a critical value.

Keywords

fractional non-linear Schrödinger equationstanding wavesconcentration-compactness principleblow-up profileperiodic potential

MSC classification

Primary: 35A15: Variational methods
Secondary: 35B44: Blow-up 35J35: Variational methods for higher-order elliptic equations 35Q55: NLS-like equations (nonlinear Schrödinger)
Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 150 , Issue 6 , December 2020 , pp. 3252 - 3292
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Copyright
Copyright © The Author(s) 2020. Published by The Royal Society of Edinburgh

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