  We consider deterministic Mean Field Games (MFG) in all Euclidean space with
a cost functional continuous with respect to the distribution of the agents and
attaining its minima in a compact set. We first show that the static MFG with
such a cost has an equilibrium, and we build from it a solution of the ergodic
MFG system of 1st order PDEs with the same cost. Next we address the long-time
limit of the solutions to finite horizon MFG with cost functional satisfying
various additional assumptions, but not the classical Lasry-Lions monotonicity
condition. Instead we assume that the cost has the same set of minima for all
measures describing the population. We prove the convergence of the
distribution of the agents and of the value function to a solution of the
ergodic MFG system as the horizon of the game tends to infinity, extending to
this class of MFG some results of weak KAM theory.
