GROUND STATE SOLUTIONS FOR THE CHERN–SIMONS–SCHRÖDINGER SYSTEM WITH HARTREE–TYPE NONLINEARITY IN $ \mathbb{R}^{2}$

In this paper, we consider the following Chern–Simons–Schrödinger system with Hartree–type nonlinearity in $\mathbb{R}^{2}$ <p class="disp_formula">$\left\{\begin{array}{l}-\Delta u+ (1+\mu V(x))u+ A_{0}u+ A_{1}^{2}u+ A_{2}^{2}u=\left(|x|^{-\alpha}*|u|^{p}\right)|u|^{p-2}u, \\\partial_{1}A_{0}=A_{2}u^{2}, ~\partial_{2}A_{0}=-A_{1}u^{2}, \\\partial_{1}A_{2}-\partial_{2}A_{1}=-\frac{1}{2}|u|^{2}, ~\partial_{1}A_{1}+\partial_{2}A_{2}=0, \end{array}\right.$ where $p&gt;3$, $\alpha\in(0, 2)$, $\mu&gt;0$ is a parameter, $V(x)$ is a nonnegative continuous potential well satisfying some conditions and $*$ is a notation for the convolution of two functions in $\mathbb{R}^{2}$. By using the Nehari manifold technique and the concentration compactness principle, we obtain the existence of ground state solutions for the above problem when the parameter $\mu$ is sufficiently large. Furthermore, the concentration behaviors of these solutions are also explored.