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Many synchronized vector solutions for a Bose–Einstein system
Published online by Cambridge University Press: 13 January 2020
Abstract
This paper is concerned with the following nonlinear Schrödinger system in ${\mathbb R}^3$
$$\left\{ {\beging{matrix}{ {-\Delta u + (1 + \alpha P(x))u = \mu u^3 + \beta uv^2,} \hfill & {x\in {\open R}^3,} \hfill \cr {-\Delta v + (1 + \alpha Q(x))u = \nu v^3 + \beta u^2v,} \hfill & {x\in {\open R}^3,} \hfill \cr {u,v > 0,} \hfill & {x\in {\open R}^3,} \hfill \cr } } \right.$$
$\beta \in {\mathbb R}$ 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.
MSC classification
- Type
- Research Article
- Information
- Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 150 , Issue 6 , December 2020 , pp. 3293 - 3320
- Copyright
- Copyright © The Author(s), 2020. Published by The Royal Society of Edinburgh
References
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