In this paper, we consider the following linearly coupled Schrödinger system: ($P_{\epsilon }$)$\left\{\begin{array}{c}-{\epsilon }^2\Delta u+u=u^3+\lambda v\text{ in }\Omega ,\hfill \\ -{\epsilon }^2\Delta v+v=v^3+\lambda u\text{ in }\Omega ,\hfill \\ u>0,v>0\text{ in }\Omega ,\hfill \\ \frac{\partial u}{\partial n}=\frac{\partial v}{\partial n}=0\text{ on }\partial \Omega ,\hfill \end{array}\right.$where $0<\epsilon <1$ is a small parameter, $0<\lambda <1$ is a coupling parameter, $\Omega$ is a smooth and bounded domain in ${\mathbb{R}}^3$, and $n$ is the outer normal vector defined on $\partial \Omega$, the boundary of $\Omega$. Motivated by the works of Ao and Wei (2014) and Ao et al. (2013), we use the Lyapunov–Schmidt reduction method to construct a positive synchronized solution of the problem ($P_{\epsilon }$) with $O\left({\epsilon }^{-3}\right)$ interior spikes for sufficiently small $\epsilon$ and some $\lambda$ near 1. In particular, we also show that the problem ($P_{\epsilon }$) has exactly $O\left({\epsilon }^{-3}\right)$ many positive synchronized solutions.

在本文中，我们考虑下面的线性耦合薛定谔系统：（$P_{\epsilon }$）$\left\{\begin{array}{c}-{\epsilon }^{\mathrm{2个}}\Delta ü+ü={ü}^3+\lambda v\text{ 在 }\Omega ，\hfill \\ -{\epsilon }^{\mathrm{2个}}\Delta v+v=v^3+\lambda ü\text{ 在 }\Omega ，\hfill \\ ü>0，v>0\text{ 在 }\Omega ，\hfill \\ \frac{\partial ü}{\partial ñ}=\frac{\partial v}{\partial ñ}=0\text{ 上 }\partial \Omega ，\hfill \end{array}\right.$在哪里 $0<\epsilon <\mathrm{1个}$ 是一个小参数， $0<\lambda <\mathrm{1个}$ 是一个耦合参数， $\Omega$ 是一个光滑且有界的区域 ${\mathbb{[R}}^3$， 和 $ñ$ 是在上定义的外部法线向量 $\partial \Omega$，的边界 $\Omega$。受到Ao和Wei（2014）以及Ao等人（2014）的启发。（2013），我们使用Lyapunov–Schmidt约简方法来构造问题的正同步解决方案（$P_{\epsilon }$） 和 $Ø（{\epsilon }^{-3}）$ 内部鞋钉足够小 $\epsilon$ 还有一些 $\lambda$ 接近1。特别是，我们还证明了这个问题（$P_{\epsilon }$）完全有 $Ø（{\epsilon }^{-3}）$ 许多积极的同步解决方案。