A variational approach to second order mean field games with density constraints: the stationary case

In this paper we study second order stationary Mean Field Game systems under density constraints on a bounded domain $\Omega \subset \mathbb{R}^d$. We show the existence of weak solutions for power-like Hamiltonians with arbitrary order of growth. Our strategy is a variational one, i.e. we obtain the Mean Field Game system as the optimality condition of a convex optimization problem, which has a solution. When the Hamiltonian has a growth of order $q' \in ]1, d/(d-1)[$, the solution of the optimization problem is continuous which implies that the problem constraints are qualified. Using this fact and the computation of the subdifferential of a convex functional introduced by Benamou-Brenier, we prove the existence of a solution of the MFG system. In the case where the Hamiltonian has a growth of order $q'\geq d/(d-1)$, the previous arguments do not apply and we prove the existence by means of an approximation argument.