Local Minimizers in Second Order Mean-field Games Systems with Choquard Coupling

TL;DR

This paper investigates the existence and characterization of solutions in second-order ergodic mean-field games with Choquard coupling, identifying conditions for global and local minimizers across different regimes.

Contribution

It provides new sufficient conditions for solutions in MFGs with Choquard coupling, including local minimizers in the supercritical regime, using variational methods.

Findings

Solutions are global minimizers in subcritical and critical regimes.

Solutions are local minimizers in supercritical regime.

The approach employs variational methods, regularization, and Hardy-Littlewood-Sobolev inequality.

Abstract

Mean-field Games systems (MFGs) serve as paradigms to describe the games among a huge number of players. In this paper, we consider the ergodic Mean-field Games systems in the bounded domain with Neumann boundary conditions and the decreasing Choquard coupling. Our results provide sufficient conditions for the existence of solutions to MFGs with Choquard-type coupling. More specifically, in the mass-subcritical and critical regimes, the solutions are characterized as global minimizers of the associated energy functional. In the case of mass supercritical exponents, up to the Sobolev critical threshold, the solutions correspond to local minimizers. The proof is based on variational methods, in which the regularization approximation, convex duality argument, elliptic regularity and Hardy-Littlewood-Sobolev inequality are comprehensively employed.