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Transactions of the American Mathematical Society

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Normalized solutions to the mixed dispersion nonlinear Schrödinger equation in the mass critical and supercritical regime


Authors: Denis Bonheure, Jean-Baptiste Casteras, Tianxiang Gou and Louis Jeanjean
Journal: Trans. Amer. Math. Soc. 372 (2019), 2167-2212
MSC (2010): Primary 35Q55, 35J30, 35J50; Secondary 35B35, 35Q60, 35Q40
DOI: https://doi.org/10.1090/tran/7769
Published electronically: April 4, 2019
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Abstract | References | Similar Articles | Additional Information

Abstract: In this paper, we study the existence of solutions to the mixed dispersion nonlinear Schrödinger equation

$\displaystyle \gamma \Delta ^2 u -\Delta u + \alpha u=\vert u\vert^{2 \sigma } u, \qquad u \in H^2({\mathbb{R}}^N), $

under the constraint

$\displaystyle \int _{{\mathbb{R}}^N}\vert u\vert^2 \, dx =c>0. $

We assume that $ \gamma >0, N \geq 1, 4 \leq \sigma N < \frac {4N}{(N-4)^+}$, whereas the parameter $ \alpha \in {\mathbb{R}}$ will appear as a Lagrange multiplier. Given $ c \in {\mathbb{R}}^+$, we consider several questions including the existence of ground states and of positive solutions and the multiplicity of radial solutions. We also discuss the stability of the standing waves of the associated dispersive equation.

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Additional Information

Denis Bonheure
Affiliation: Département de Mathématiques, Université Libre de Bruxelles, C.P. 214, Boulevard du triomphe, B-1050 Bruxelles, Belgium
Email: Denis.Bonheure@ulb.ac.be

Jean-Baptiste Casteras
Affiliation: Département de Mathématiques, Université Libre de Bruxelles, C.P. 214, Boulevard du triomphe, B-1050 Bruxelles, Belgium
Email: jeanbaptiste.casteras@gmail.com

Tianxiang Gou
Affiliation: Laboratoire de Mathématiques (UMR 6623), Université Bourgogne Franche-Comté, 16 Route de Gray, 25030 Besançon Cedex, France; and School of Mathematics and Statistics, Lanzhou University, Lanzhou, Gansu 730000, People’s Republic of China
Email: gou.tianxiang@gmail.com

Louis Jeanjean
Affiliation: Laboratoire de Mathématiques (UMR 6623), Université Bourgogne Franche-Comté, 16 Route de Gray, 25030 Besançon Cedex, France
Email: louis.jeanjean@univ-fcomte.fr

DOI: https://doi.org/10.1090/tran/7769
Received by editor(s): February 24, 2018
Received by editor(s) in revised form: September 13, 2018, and October 3, 2018
Published electronically: April 4, 2019
Additional Notes: The first author was supported by MIS F.4508.14 (FNRS) and PDR T.1110.14F (FNRS). He was also partially supported by the project ERC Advanced Grant 2013 no. 339958: “Complex Patterns for Strongly Interacting Dynamical Systems—COMPAT” and by ARC AUWB-2012-12/17-ULB1- IAPAS
The second author was supported by MIS F.4508.14 (FNRS) and PDR T.1110.14F (FNRS)
This work has been carried out in the framework of the project NONLOCAL (ANR-14-CE25-0013), funded by the French National Research Agency (ANR)
Article copyright: © Copyright 2019 American Mathematical Society