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Many synchronized vector solutions for a Bose–Einstein system

Part of: Elliptic equations and systems

Published online by Cambridge University Press:  13 January 2020

Wei Long ,
Zhongwei Tang  and
Sudan Yang
Wei Long
Affiliation:
College of Mathematics and Information Science, Jiangxi Normal University, Nanchang, Jiangxi 330022, P. R. China (lwhope@jxnu.edu.cn; 2353692672@qq.com)
Zhongwei Tang
Affiliation:
School of Mathematical Sciences, Beijing Normal University, Beijing100875, P. R. China (tangzw@bnu.edu.cn)
Sudan Yang
Affiliation:
College of Mathematics and Information Science, Jiangxi Normal University, Nanchang, Jiangxi 330022, P. R. China (lwhope@jxnu.edu.cn; 2353692672@qq.com)
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Abstract

This paper is concerned with the following nonlinear Schrödinger system in ${\mathbb R}^3$

$$\left\{ {\beging{matrix}{ {-\Delta u + (1 + \alpha P(x))u = \mu u^3 + \beta uv^2,} \hfill & {x\in {\open R}^3,} \hfill \cr {-\Delta v + (1 + \alpha Q(x))u = \nu v^3 + \beta u^2v,} \hfill & {x\in {\open R}^3,} \hfill \cr {u,v > 0,} \hfill & {x\in {\open R}^3,} \hfill \cr } } \right.$$
where $\beta \in {\mathbb R}$ is a coupling constant, $\mu ,\nu $ are positive constants, P,Q are weight functions decaying exponentially to zero at infinity, α can be regarded as a parameter. This type of system arises, in particular, in models in Bose–Einstein condensates theory and Kerr-like photo refractive media.

We prove that, for any positive integer k > 1, there exists a suitable range of α such that the above problem has a non-radial positive solution with exactly k maximum points which tend to infinity as $\alpha \to +\infty $ (or $0^+$). Moreover, we also construct prescribed number of sign-changing solutions.

Keywords

multiple positive solutionsnonlinear Bose–Einstein system

MSC classification

Primary: 35J20: Variational methods for second-order elliptic equations
Secondary: 35J60: Nonlinear elliptic equations
Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 150 , Issue 6 , December 2020 , pp. 3293 - 3320
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Copyright
Copyright © The Author(s), 2020. Published by The Royal Society of Edinburgh

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