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In this paper, we study the following coupled Choquard system in $\mathbbR^N$: $$\left\{\begin{align}&-\Delta u+A(x)u=\frac{2p}{p+q} \bigl(I_\alpha\ast|v|^q\bigr)|u|^{p-2}u,\\ &-\Deltav+B(x)v=\frac{2q}{p+q}\bigl(I_\alpha\ast|u|^p\bigr)|v|^{q-2}v,\\ &\ u(x)\to0\ \\hbox{and}\ \ v(x)\to0\ \ \hbox{as}\ |x|\to\infty,\end{align}\right.$$ where$\alpha\in(0,N)$ and $\frac{N+\alpha}{N}<p,\ q<2_*^\alpha$, in which$2_*^\alpha$ denotes $\frac{N+\alpha}{N-2}$ if $N\geq 3$ and $2_*^\alpha :=\infty$ if $N=1,\ 2$. The function $I_\alpha$ is a Riesz potential. By usingNehari manifold method, we obtain the existence of positive ground statesolution 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 $p\neq q$ and $u(x)\neq v(x)$. Moreoverit seems to be the first existence result for the case of $p\neq q$.

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