We provide the existence of classical solutions to stationary mean-field game systems in the
whole space ,
with coercive potential and aggregating local coupling, under general conditions on
the Hamiltonian. The only structural assumption we make is on the growth at
infinity of the coupling term in terms of the growth of the Hamiltonian. This result is
obtained using a variational approach based on the analysis of the nonconvex energy
associated to the system. Finally, we show that in the vanishing viscosity limit, mass
concentrates around the flattest minima of the potential. We also describe the
asymptotic shape of the rescaled solutions in the vanishing viscosity limit, in
particular proving the existence of ground states, i.e., classical solutions to mean-field
game systems in the whole space without potential, and with aggregating
coupling.
