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Non-degeneracy of synchronized vector solutions for weakly coupled nonlinear schrödiner systems

Part of: Elliptic equations and systems Partial differential equations

Published online by Cambridge University Press:  22 April 2022

Qing Guo  and
Leiga Zhao
Qing Guo
Affiliation:
College of Science, Minzu University of China, Beijing 100081, China (guoqing0117@163.com)
Leiga Zhao
Affiliation:
School of Mathematics and Statistics, Beijing Technology and Business University, (zhaoleiga@163.com)
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Abstract

In this paper, we consider the following two-component Schrödiner system

\[ \begin{cases} -\Delta u_1 +V_1(x) u_1= \mu_1 u_1^{3} +\beta u_1u_2^{2} & \text{in}\;\mathbb{R}^{3},\\ -\Delta u_2+V_2(x) u_2= \mu_2 u_2^{3} +\beta u_1^{2}u_2 & \text{in}\;\mathbb{R}^{3}. \end{cases} \]
First, we revisit the proof of the existence of an unbounded sequence of non-radial positive vector solutions of synchronized type obtained in S. Peng and Z. Wang [Segregated and synchronized vector solutions for nonlinear Schrödinger systems, Arch. Rational Mech. Anal. 208 (2013), 305–339] to give a point-wise estimate of the solutions. Taking advantage of these estimates, we then show a non-degeneracy result of the synchronized solutions in some suitable symmetric space by use of the locally Pohozaev identities. The main difficulties of BEC systems come from the interspecies interaction between the components, which never appear in the study of single equations. The idea used to estimate the coupling terms is inspired by the characterization of the Fermat points in the famous Fermat problem, which is the main novelty of this paper.

Keywords

Schrödinger systemsNon-degeneracyPopozaev identitiesLyapunov–Schmidt reductionSynchronized solutions

MSC classification

Primary: 35B25: Singular perturbations
Secondary: 35J47: Second-order elliptic systems 35J60: Nonlinear elliptic equations
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 65 , Issue 2 , May 2022 , pp. 441 - 459
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Copyright
Copyright © The Author(s), 2022. Published by Cambridge University Press on Behalf of The Edinburgh Mathematical Society

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