Ergodic Mean-Field Games with aggregation of Choquard-type

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