Existence of ground state solutions for the modified Chern-Simons-Schrödinger equations with general Choquard type nonlinearity

In this paper, we are concerned with the following modified Schrödinger equation

$$\begin{array}{l}-\mathrm{\Delta }u+V(|x|)u-\kappa u\mathrm{\Delta }(u^2)+\\ q\frac{h^2(|x|)}{|x{|}^2}(1+\kappa u^2)u+q\left({\int }_{|x|}^{+\mathrm{\infty }}\frac{h(s)}{s}(2+\kappa u^2(s))u^2(s)\mathrm{d}s\right)u=(I_{\alpha }\ast F(u))f(u),x\in {\mathbb{R}}^2,\end{array}$$

where $\kappa$, $q>0$, $I_{\alpha }$ is a Riesz potential, $\alpha \in (0,2)$ and $V\in \mathcal{C}({\mathbb{R}}^2,\mathbb{R})$, $F(t)={\int }_0^{t}f(s)\mathrm{d}s$. Under appropriate assumptions on $f$ and $V(x)$, by using the variational methods, we establish the existence of ground state solutions of the above equation.