Multiple positive normalized solutions for nonlinear Schrödinger systems

By Tianxiang Gou, Louis Jeanjean. 2017

145 citations recorded by the source corpus.

We consider the existence of multiple positive solutions to the nonlinear Schrödinger systems set on , under the constraint Here are prescribed, , and the frequencies are unknown and will appear as Lagrange multipliers. Two cases are studied, the first when , the second when In both cases, assuming that is sufficiently small, we prove the existence of two positive solutions. The first one is a local minimizer for which we establish the compactness of the minimizing sequences and also discuss the orbital stability of the associated standing waves. The second solution is obtained through a constrained mountain pass and a constrained linking respectively.

Subfields: Applied Mathematics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/0a527fa00b935e45788580849f6165b54c5e8bc3

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In Applied Mathematics

In Mathematical Physics

Citations in the corpus

Normalized ground states for the NLS equation with combined nonlinearities: The Sobolev critical case (2019, unknown): “More recently, this kind of idea has been exploited and further developed in other contexts, and more in general the search for normalized solutions in cases when the energy is unbounded from below on the L-constraint attracted much attention: we refer to [1, 7, 12, 13, 16, 20, 34] for normalized so”
Normalized solutions for a Schrödinger equation with critical growth in RN\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{documen (2021, unknown): “For the reader interested in normalized solutions for the Schrödinger equations, we would also like to refer [9,11,12,17,25,29,31,34,38,39,41] and references therein.”
Orbital stability of ground states for a Sobolev critical Schrödinger equation (2020, background): “Different strategies have been recently implemented to deal with this issue [4, 20,34], all relying on a suitable choice of the set where the local minima are searched.”
Existence and asymptotic behavior of normalized ground states for Sobolev critical Schrödinger systems (2022, methodology): “We refer the reader to [7, 26, 27, 28, 29, 30, 39, 40, 43] for scalar equations, [8, 9, 10, 11, 14, 21, 22, 31] for systems of two equations, [35] for systems of k equations.”
Normalized solutions to mass supercritical Schrödinger equations with negative potential (2021, background): “After [22], only recently more general autonomous equations and systems have been considered, also refining and developing this initial strategy, see [4, 6, 8, 5, 17, 9, 21, 34, 35] and references therein.”
Multiple normalized solutions for a Sobolev critical Schrödinger equation (2020, unknown): “See, for example, [8, 18, 21, 34].”
Normalized ground states for the NLS equation with combined nonlinearities (2018, unknown): “More recently, however, problems of this type received much attention, see [1, 10–12, 15, 16, 32] for normalized solutions to scalar equations in the whole space R , [6–9, 28] for normalized solutions to systems”