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.

中文翻译：

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Bose-Einstein 系统的许多同步矢量解

本文关注以下非线性薛定谔系统${\mathbb R}^3$$$\left\{ {\beging{matrix}{ {-\Delta u + (1 + \alpha P(x))u = \mu u^3 + \beta uv^2,} \hfill & {x\在 {\open R}^3,} \hfill \cr {-\Delta v + (1 + \alpha Q(x))u = \nu v^3 + \beta u^2v,} \hfill & {x \in {\open R}^3,} \hfill \cr {u,v > 0,} \hfill & {x\in {\open R}^3,} \hfill \cr } } \right.$$在哪里$\beta \in {\mathbb R}$是耦合常数，$\亩 ,\nu $是正常数，磷,问权重函数在无穷远处呈指数衰减到零，α可以看作一个参数。这种类型的系统尤其出现在玻色-爱因斯坦凝聚理论和类克尔光折射介质的模型中。我们证明，对于任何正整数ķ> 1，存在一个合适的范围α使得上述问题有一个非径向正解ķ趋于无穷大的最大点为$\alpha \to +\infty $（要么$0^+$）。此外，我们还构建了规定数量的符号改变解决方案。