  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.
