Existence for stationary mean field games with quadratic Hamiltonians with congestion

In this paper, we investigate the existence and uniqueness of solutions to a stationary mean field game model introduced by J.-M. Lasry and P.-L. Lions. This model features a quadratic Hamiltonian with possibly singular congestion effects. Thanks to a new class of a-priori bounds, combined with the continuation method, we prove the existence of smooth solutions in arbitrary dimensions.