Existence of positive solutions for an approximation of stationary mean-field games

TL;DR

This paper proves the existence of positive solutions for a regularized stationary mean-field game model with low-order regularization, providing both theoretical insights and practical advantages for numerical implementation.

Contribution

It introduces a low-order regularization approach for stationary mean-field games and establishes existence results using a combination of a priori estimates and the continuation method.

Findings

Existence of positive density solutions proven

Low-order regularizations are easier to implement numerically

Provides a theoretical foundation for low-order regularized models

Abstract

Here, we consider a regularized mean-field game model that features a low-order regularization. We prove the existence of solutions with positive density. To do so, we combine a priori estimates with the continuation method. In contrast with high-order regularizations, the low-order regularizations are easier to implement numerically. Moreover, our methods give a theoretical foundation for this approach.