Ergodic Mean Field Games: existence of local minimizers up to the Sobolev critical case

TL;DR

This paper studies the existence of solutions to viscous ergodic Mean Field Games systems with Neumann boundary conditions, identifying conditions for global and local minimizers across different growth regimes, including the Sobolev critical case.

Contribution

It introduces a variational approach to establish the existence of local minimizers in the supercritical case, extending results up to the Sobolev critical threshold.

Findings

Existence of global minimizers in subcritical and critical cases

Existence of local minimizers in supercritical case

Extension of results up to Sobolev critical case

Abstract

We investigate the existence of solutions to viscous ergodic Mean Field Games systems in bounded domains with Neumann boundary conditions and local, possibly aggregative couplings. In particular we exploit the associated variational structure and search for constrained minimizers of a suitable functional. Depending on the growth of the coupling, we detect the existence of global minimizers in the mass subcritical and critical case, and of local minimizers in the mass supercritical case, notably up to the Sobolev critical case.