Existence of positive ground state solutions for the coupled Choquard system with potential

In this paper, we study the following coupled Choquard system in ${\mathbb{R}}^N$: where $\alpha \in (0,N)$ and , in which $2_{\ast }^{\alpha }$ denotes $\frac{N+\alpha }{N-2}$ if $N\ge 3$ and $2_{\ast }^{\alpha }:=\infty$ if $N=1,2$. The function $I_{\alpha }$ is a Riesz potential. By using Nehari manifold method, we obtain the existence of a positive ground state solution in the case of bounded potential and periodic potential, respectively. In particular, the nonlinear term includes the well-studied case $p=q$ and $u(x)=v(x)$, and the less-studied case $p\ne q$ and $u(x)\ne v(x)$. Moreover, it seems to be the first existence result for the case $p\ne q$.