# Ground states and concentration of mass in stationary Mean Field Games with superlinear Hamiltonians

created by cesaroni on 07 Feb 2018
modified on 14 Feb 2018

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Submitted Paper

Inserted: 7 feb 2018
Last Updated: 14 feb 2018

Year: 2018

Abstract:

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.

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