Mass concentration for Ergodic Choquard Mean-Field Games

We study concentration phenomena in the vanishing viscosity limit for second-order stationary Mean-Field Games systems defined in the whole space ℝN with Riesz-type aggregating coupling and external confining potential. In this setting, every player of the game is attracted toward congested areas and the external potential discourages agents from being far away from the origin. Assuming some restrictions on the strength of the attractive nonlocal term depending on the growth of the Hamiltonian, we study the asymptotic behavior of solutions in the vanishing viscosity limit. First, we obtain existence of classical solutions to potential-free MFG systems with Riesz-type coupling. Secondly, we prove concentration of mass around minima of the potential.