In this paper we study second order stationary Mean Field Game systems under density constraints on a bounded domain . 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 , 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 and Brenier (see ), we prove the existence of a solution of the MFG system. In the case where the Hamiltonian has a growth of order , the previous arguments do not apply and we prove the existence by means of an approximation argument.
Mészáros, A. R., & Silva, F. J. (2015). A variational approach to second order mean field games with density constraints: The stationary case. Journal de Mathématiques Pures et Appliquées, 104(6), 1135-1159. https://doi.org/10.1016/j.matpur.2015.07.008