# Concentration of ground states in stationary Mean-Field Games systems

created by cesaroni on 30 May 2017
modified on 16 Aug 2017

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Submitted Paper

Inserted: 30 may 2017
Last Updated: 16 aug 2017

Year: 2017

Abstract:

In this paper we provide the existence of classical solutions to some stationary mean field game systems in the whole space $\mathbb{R}^N$, with coercive potential and aggregating local coupling. This result is obtained under structural assumptions on the growth at infinity of the coupling term and of the Hamiltonian, by using a variational approach based on the analysis of the non-convex energy associated to the system. Moreover, we analyse the limit of the system as the diffusion vanishes, and we show that phenomena of concentration of mass appear. 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.