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.

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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：$$ \ begin {aligned} \ left \ {\ begin {array} {l}-\ triangle u + \ mu u ^ {3} = \ lambda u + \ kappa v + \ beta uv ^ {2} \ \ \ text {in} \ \ \ Omega，\\-\\ triangle v + \ mu v ^ {3} = \ lambda v + \ kappa u + \ beta vu ^ {2} \ \ \ text {in} \ \ \ Omega，\\ u，v> 0 \ \ text {in} \ \ Omega，\ u = v = 0 \ \ qquad \，\，\，\，\ text {on} \ \ partial \ Omega，\ end {array} \ right。\ end {aligned} $$和$$ \ begin {aligned} \ left \ {\ begin {array} {l}-\ triangle u + \ lambda u + \ kappa v = \ mu u ^ {3} + \ beta uv ^ {2} \ \ \ text { in} \ \ \ Omega，\\-\ triangle v + \ lambda v + \ kappa u = \ mu v ^ {3} + \ beta vu ^ {2} \ \ \ text {in} \ \ \ Omega，\\ u ，v> 0 \ \ text {in} \ \ Omega，\ u = v = 0 \ \ qquad \ quad \ text {on} \ \ partial \ Omega \ end {array} \ right。\ end {aligned} $$在\（\ lambda \）和耦合常数\（\ kappa \），\（\ beta \）的范围内，其中\（\ Omega \）是\（\ {\ 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。