  We consider second-order ergodic Mean-Field Games systems in the whole space
$\mathbb{R}^N$ with coercive potential and aggregating nonlocal coupling,
defined in terms of a Riesz interaction kernel. These MFG systems describe Nash
equilibria of games with a large population of indistinguishable rational
players attracted toward regions where the population is highly distributed.
Equilibria solve a system of PDEs where an Hamilton-Jacobi-Bellman equation is
combined with a Kolmogorov-Fokker-Planck equation for the mass distribution.
Due to the interplay between the strength of the attractive term and the
behavior of the diffusive part, we will obtain three different regimes for the
existence and non existence of classical solutions to the MFG system. By means
of a Pohozaev-type identity, we prove nonexistence of regular solutions to the
MFG system without potential in the Hardy-Littlewood-Sobolev-supercritical
regime. On the other hand, using a fixed point argument, we show existence of
classical solutions in the Hardy-Littlewood-Sobolev-subcritical regime at least
for masses smaller than a given threshold value. In the mass-subcritical regime
we show that actually this threshold can be taken to be $+\infty$.
