Existence and multiplicity of solutions to the mean-field games model with mixed interactions

TL;DR

This paper develops a minimization method on the Pohozaev manifold to establish existence and multiplicity of solutions for stationary mean-field games with mixed interactions and homogeneous Hamiltonians.

Contribution

It introduces a novel minimization approach on the Pohozaev manifold for stationary MFG models with 2-homogeneous Hamiltonians and extends results to general p-homogeneous cases.

Findings

Proved existence of solutions for stationary MFG models with mixed interactions.

Established multiplicity of radial solutions under general conditions.

Extended techniques to broader classes of Hamiltonians beyond previous work.

Abstract

In this paper, we consider the stationary version of the Mean-Field Games (MFG) models. Inspired by \cite{Albuquerque-Silva2020, Bieganowski-Mederski2021, Lin-Wei05, Mederski-Schino2021}, we develop the minimization method on the Pohozaev manifold introduced in \cite{Soave20JDE, Soave20JFA} for the existence theory of the stationary version of the Mean-Field Games (MFG) models with $2$-homogeneous hamiltonians and mixed interactions. As applications, we prove the existence and multiplicity of radial solutions of the Mean-Field Games (MFG) models with general $p$-homogeneous hamiltonians and mixed interactions under more general conditions, some of which are even new for $2$-homogeneous hamiltonians. We hope that our techniques and ideas introduced in this paper would be helpful in understanding the optimal value of the total mass in the existence theory of radial solutions to the Mean-Field Games (MFG) models with general $p$-homogeneous hamiltonians and mixed interactions, as well as that of other models.