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Normalized solutions for nonlinear Schrödinger systems

Part of: Equations and inequalities involving nonlinear operators Elliptic equations and systems Equations of mathematical physics and other areas of application Variational problems in infinite-dimensional spaces Partial differential equations Spectral theory and eigenvalue problems

Published online by Cambridge University Press:  20 November 2017

Thomas Bartsch  and
Louis Jeanjean
Thomas Bartsch
Affiliation:
Mathematisches Institut, Universität Giessen, Arndtstraße 2, 35392 Giessen, Germany ([email protected])
Louis Jeanjean
Affiliation:
Laboratoire de Mathématiques (UMR 6623), Université Bourgogne Franche-Comté, 16 Route de Gray, 25030 Besançon Cedex, France ([email protected])
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Abstract

We consider the existence of normalized solutions in H1(ℝN) × H1(ℝN) for systems of nonlinear Schr¨odinger equations, which appear in models for binary mixtures of ultracold quantum gases. Making a solitary wave ansatz, one is led to coupled systems of elliptic equations of the form

and we are looking for solutions satisfying

where a1> 0 and a2> 0 are prescribed. In the system, λ1 and λ2 are unknown and will appear as Lagrange multipliers. We treat the case of homogeneous nonlinearities, i.e. , with positive constants β, μi, pi, ri. The exponents are Sobolev subcritical but may be L2-supercritical. Our main result deals with the case in which in dimensions 2 ≤ N ≤ 4. We also consider the cases in which all of these numbers are less than 2 + 4/N or all are bigger than 2 + 4/N.

Keywords

nonlinear Schrödinger systemssolitary wavesnormalized solutionsvariational methodsconstrained linking

MSC classification

Secondary: 35J50: Variational methods for elliptic systems 35B08: Entire solutions 35J47: Second-order elliptic systems 35P30: Nonlinear eigenvalue problems, nonlinear spectral theory 35Q55: NLS-like equations (nonlinear Schrödinger) 47J30: Variational methods 58E05: Abstract critical point theory (Morse theory, Ljusternik-Schnirelman (Lyusternik-Shnirel'man) theory, etc.)
Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 148 , Issue 2 , April 2018 , pp. 225 - 242
Copyright
Copyright © Royal Society of Edinburgh 2018 

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