In this work we study second order stationary Mean Field Game systems under density constraints. We show the existence of weak solutions for power-like Hamiltonians with arbitrary order of growth. The density constraint introduces a natural pressure in the system. Our strategy is a variational one, which consists in the analysis of two optimal control problems in duality. The first problem has its roots in the so-called dynamic or Benamou-Brenier formulation of the Monge-Kantorovich optimal transportation problem. In the case of sub-quadratic Hamiltonians the proof is carried out using classical tools from convex analysis. In the super-quadratic case we prove the result by means of an approximation argument. This is a joint work with A. R. Mészáros (Université d'Orsay).