For the Choquard equation, which is a nonlocal nonlinear Schrödinger type equation,
$$-\Delta u+V_{\mu, \nu} u=(I_\alpha\ast \vert u \vert ^{({N+\alpha})/{N}}){ \vert u \vert }^{{\alpha}/{N}-1}u,\quad {\rm in} \ {\open R}^N, $$ where
$N\ges 3$,
Vμ,ν :ℝ
N → ℝ is an external potential defined for μ, ν > 0 and
x ∈ ℝ
N by
Vμ,ν(
x) = 1 − μ/(ν
2 + |
x|
2) and
$I_\alpha : {\open R}^N \to 0$ is the Riesz potential for α ∈ (0,
N), we exhibit two thresholds μ
ν, μ
ν > 0 such that the equation admits a positive ground state solution if and only if μ
ν < μ < μ
ν and no ground state solution exists for μ < μ
ν. Moreover, if μ > max{μ
ν,
N2(
N − 2)/4(
N + 1)}, then equation still admits a sign changing ground state solution provided
$N \ges 4$ or in dimension
N = 3 if in addition 3/2 < α < 3 and
$\ker (-\Delta + V_{\mu ,\nu }) = \{ 0\} $, namely in the non-resonant case.