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

## TL;DR

This paper proves the existence of classical solutions for stationary mean field game systems in unbounded space with coercive potential and aggregating coupling, and analyzes the mass concentration in the vanishing viscosity limit.

## Contribution

It introduces a variational method to establish solutions under general conditions and describes the asymptotic behavior of solutions as viscosity vanishes.

## Key findings

- Existence of classical solutions in ^N with coercive potential.
- Mass concentrates around the flattest minima in the vanishing viscosity limit.
- Existence of ground states without potential in the MFG system.

## 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 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.