# On the variational formulation of some stationary second order mean field games systems

## TL;DR

This paper develops a variational method to prove the existence of solutions for second order stationary mean field games with Neumann boundary conditions, including cases with density constraints and multiple populations.

## Contribution

It introduces a general variational framework that handles complex coupling terms, density constraints, and multi-population systems in mean field games.

## Key findings

- Proves existence of solutions under broad conditions.
- Extends previous results to systems with density constraints.
- Applicable to multi-population mean field game systems.

## Abstract

We consider the variational approach to prove the existence of solutions of second order stationary Mean Field Games on a bounded domain $\Omega\subseteq \mathbb{R}^{d}$, with Neumann boundary conditions, and with and without density constraints. We consider Hamiltonians which growth as $|\cdot|^{q'}$, where $q'=q/(q-1)$ and $q>d$. Despite this restriction, our approach allows us to prove the existence of solutions in the case of rather general coupling terms. When density constraints are taken into account, our results improve those in \cite{MesSil}. Furthermore, our approach can be used to obtain solutions of systems with multiple populations.