  In this paper we provide the existence of classical solutions to stationary
mean field game systems in the whole space $\mathbb{R}^N$, 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
non-convex 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.
