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In this paper we study second order stationary Mean Field Game systems underdensity constraints on a bounded domain $\Omega \subset \mathbb{R}^d$. We showthe existence of weak solutions for power-like Hamiltonians with arbitraryorder of growth. Our strategy is a variational one, i.e. we obtain the MeanField 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 impliesthat the problem constraints are qualified. Using this fact and the computationof the subdifferential of a convex functional introduced by Benamou-Brenier, weprove the existence of a solution of the MFG system. In the case where theHamiltonian has a growth of order $q'\geq d/(d-1)$, the previous arguments donot apply and we prove the existence by means of an approximation argument.

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