Existence and local uniqueness of multi-peak solutions for the Chern-Simons-Schrödinger system

In the present paper, we consider the Chern-Simons-Schrödinger system \begin{equation} \left{ \begin{aligned} &-\varepsilon^{2}Δu+V(x)u+(A_{0}+A_{1}^{2}+A_{2}^{2})u=|u|^{p-2}u,,,,,x\in $\mathbb{R}$^2,\ &\partial_1 A_0 = A_2 u^2,\ \partial_{2}A_{0}=-A_{1}u^{2},\ &\partial_{1}A_{2}-\partial_{2}A_{1}=-\frac{1}{2}|u|^{2},\ \partial_{1}A_{1}+\partial_{2}A_{2}=0,\ \end{aligned} \right. \end{equation} where $p>2,$ $\epsilon >0$ is a parameter and $V:{\mathbb{R}}^2\to \mathbb{R}$ is a bounded continuous function. Under some mild assumptions on $V(x)$, we show the existence and local uniqueness of positive multi-peak solutions. Our methods mainly use the finite dimensional reduction method, various local Pohozaev identities, blow-up analysis and the maximum principle. Because of the nonlocal terms involved by $A_0,A_1$ and $A_2,$ we have to obtain a series of new and technical estimates.