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Distribution of positive solutions to Schrödinger systems with linear and nonlinear couplings
Journal of Fixed Point Theory and Applications ( IF 1.4 ) Pub Date : 2020-03-16 , DOI: 10.1007/s11784-020-0767-y
Xinqiu Zhang , Zhitao Zhang

In this paper, we study the existence, nonexistence and uniqueness of positive solutions for the following two Schrödinger systems with linear and nonlinear couplings which arise from Bose–Einstein condensates:$$\begin{aligned} \left\{ \begin{array}{l} -\triangle u+\mu u^{3}=\lambda u+\kappa v+\beta uv^{2} \ \ \text {in} \ \ \Omega ,\\ -\triangle v+\mu v^{3}=\lambda v+\kappa u+\beta vu^{2} \ \ \text {in} \ \ \Omega ,\\ u,v>0 \ \text {in} \ \Omega , \ u=v=0 \ \qquad \,\,\,\, \text {on} \ \partial \Omega , \end{array}\right. \end{aligned}$$and$$\begin{aligned} \left\{ \begin{array}{l} -\triangle u+\lambda u+\kappa v=\mu u^{3}+\beta uv^{2} \ \ \text {in} \ \ \Omega ,\\ -\triangle v+\lambda v+\kappa u=\mu v^{3}+\beta vu^{2} \ \ \text {in} \ \ \Omega ,\\ u,v>0 \ \text {in} \ \Omega , \ u=v=0 \ \qquad \quad \text {on} \ \partial \Omega \end{array}\right. \end{aligned}$$on the range of \(\lambda \) and the coupling constants \(\kappa \), \(\beta \), where \(\Omega \) is a bounded smooth domain in \({\mathbb {R}}^{N}\), \(N\ge 1\), \(\lambda , \mu >0\). We obtain some interesting positive solutions distribution theorems in the \(\kappa \lambda \)-plane for fixed \(\beta \) in different ranges. Especially we get some uniqueness results via synchronized solution techniques.

中文翻译:

具有线性和非线性耦合的Schrödinger系统的正解的分布

在本文中,我们研究了以下两个由Bose-Einstein凝聚引起的线性和非线性耦合Schrödinger系统正解的存在性,不存在性和唯一性:$$ \ begin {aligned} \ left \ {\ begin {array} {l}-\ triangle u + \ mu u ^ {3} = \ lambda u + \ kappa v + \ beta uv ^ {2} \ \ \ text {in} \ \ \ Omega,\\-\\ triangle v + \ mu v ^ {3} = \ lambda v + \ kappa u + \ beta vu ^ {2} \ \ \ text {in} \ \ \ Omega,\\ u,v> 0 \ \ text {in} \ \ Omega,\ u = v = 0 \ \ qquad \,\,\,\,\ text {on} \ \ partial \ Omega,\ end {array} \ right。\ end {aligned} $$$$ \ begin {aligned} \ left \ {\ begin {array} {l}-\ triangle u + \ lambda u + \ kappa v = \ mu u ^ {3} + \ beta uv ^ {2} \ \ \ text { in} \ \ \ Omega,\\-\ triangle v + \ lambda v + \ kappa u = \ mu v ^ {3} + \ beta vu ^ {2} \ \ \ text {in} \ \ \ Omega,\\ u ,v> 0 \ \ text {in} \ \ Omega,\ u = v = 0 \ \ qquad \ quad \ text {on} \ \ partial \ Omega \ end {array} \ right。\ end {aligned} $$\(\ lambda \)和耦合常数\(\ kappa \)\(\ beta \)的范围内,其中\(\ Omega \)\(\ {\ mathbb {R}} ^ {N} \)\(N \ ge 1 \)\(\ lambda,\ mu> 0 \)。对于固定的\(\ kappa \ lambda \)平面,我们获得了一些有趣的正解分布定理\(\ beta \)在不同的范围内。特别是,我们通过同步解决方案技术获得了一些唯一性结果。
更新日期:2020-03-16
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