A perturbation result for system of Schrödinger equations of Bose-Einstein condensates in $\mathbb{R}^3$

Kui  Li; Zhitao Zhang

doi:10.3934/dcds.2016.36.851

Article Contents

2016, Volume 36, Issue 2: 851-860. Doi: 10.3934/dcds.2016.36.851

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A perturbation result for system of Schrödinger equations of Bose-Einstein condensates in $\mathbb{R}^3$

Kui Li^1, and
Zhitao Zhang^2,

1.
School of Mathematics and Statistics, Zhengzhou University, Zhengzhou, Henan 450001
2.
Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing, 100190

Received: January 31, 2014

Revised: January 31, 2015

Published: May 2017

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Abstract

We are concerned with the existence of positive solutions for a coupled Schrödinger system \begin{equation*} \left\{ \begin{aligned} &-\Delta{u}_1+{\lambda}_1 {u}_1={\mu}_1 {u}_1^3+\varepsilon \beta(x) {u}_1 {u}_2^2 & ~~in &~~~~ \mathbb{R}^3,\\ &-\Delta{u}_2+{\lambda}_2 {u}_2={\mu}_2 {u}_2^3+\varepsilon \beta(x) {u}_1^2 {u}_2 & ~~in & ~~~~\mathbb{R}^3,\\ &{u}_1>0, ~~{u}_2>0& ~~in & ~~~~\mathbb{R}^3,\\ &{u}_1\in H^1(\mathbb{R}^3),~~{u}_2\in H^1(\mathbb{R}^3), \end{aligned} \right. \end{equation*} where ${\lambda}_1,{\lambda}_2,{\mu}_1,{\mu}_2$ are positive constants. We use perturbation methods to prove that if $\beta \in L^r(\mathbb{R}^3)(r\geq 3)$ doesn't change sign, as corresponding $\varepsilon$ is sufficiently small the system has a positive solution of which both components are positive. Our results is also true for domain $\mathbb{R}^{2}$ and for domain $ \mathbb {R}^{N}, N \geq 4 $ when the similar system is subcritical.

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Mathematics Subject Classification: Primary: 35B20; Secondary: 35J50.

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