  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, aggregating local coupling, and under general conditions on the
Hamiltonian, completing the analysis started in the companion paper [6]. 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, and that the
asymptotic shape of the solutions in a suitable rescaled setting converges to a
ground state, i.e. a classical solution to a mean field game system without
potential.
