Abstract: In this paper, we consider the semilinear elliptic equation systems $\left\{\begin{array}{ll} -\Delta u+u=\alpha Q_{n}(x)|u|^{\alpha-2}|v|^{\beta}u &\mbox{in}\hspace{1.14mm} \mathbb{R}^{N},\\ -\Delta v+v=\beta Q_{n}(x)|u|^{\alpha}|v|^{\beta-2}v &\mbox{in}\hspace{1.14mm} \mathbb{R}^{N}, \end{array} \right.$ where $N\geqslant 3$, $\alpha$, $\beta>1$, $\alpha+\beta<2^{*}$, $2^{*}=\frac{2N}{N-2}$ and $Q_{n}$ are bounded given functions whose self-focusing cores $\{x\in\mathbb{R}^N|Q_n(x)>0\}$ shrink to a set with finitely many points as $n\rightarrow\infty$. Motivated by the work of Fang and Wang [13], we use variational methods to study the limiting profile of ground state solutions which are concentrated at one point of the set with finitely many points, and we build the localized concentrated bound state solutions for the above equation systems.

Key words: Schrödinger system, ground states solutions, bound state solutions