  For the Choquard equation, which is a nonlocal nonlinear Schr\"odinger type
equation, $ -\Delta u+V_{\mu,\nu} u=(I_\alpha\ast
|u|^{\frac{N+\alpha}{N}}){|u|}^{\frac{\alpha}{N}-1}u$, in $\mathbb{R}^N$ where
$N\ge 3$, $V_{\mu, \nu} : \mathbb{R}^N \to \mathbb{R}$ is an external potential
defined for $\mu,\nu\in\mathbb{R}$ and $x \in \mathbb{R}^N$ by $V_{\mu, \nu}
(x)=1-\mu/(\nu^2 + |x|^2)$ and $I_\alpha : \mathbb{R}^N \to 0$ is the Riesz
potential for $\alpha\in (0,N)$, we exhibit two thresholds
$\mu_{\nu},\mu^{\nu}>0$ such that the equation admits a positive ground state
solution if and only if $\mu_{\nu}<\mu<\mu^{\nu}$ and no ground state solution
exists for $\mu<\mu_{\nu}$. Moreover, if
$\mu>\max\left\{\mu^{\nu},\frac{N^2(N-2)}{4(N+1)}\right\}$, then equation still
admits a sign changing ground state solution provided $N\ge4$ or in dimension
$N=3$ if in addition $\frac{3}{2}<\alpha<3$ and ker$(-\Delta + V_{\mu, \nu}) =
\{0\}$, namely in the non-resonant case.
