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

TL;DR

This paper establishes the existence of classical solutions for stationary mean field game systems with superlinear Hamiltonians in the whole space, analyzing mass concentration and asymptotic behavior as viscosity vanishes.

Contribution

It provides a variational method to prove existence of solutions under general conditions and describes the mass concentration and limiting profiles in the vanishing viscosity limit.

Findings

Solutions concentrate around potential minima as viscosity vanishes

Solutions' asymptotic shape converges to ground states without potential

Existence results extend previous analyses to more general Hamiltonians

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.