Abstract
In this paper we establish a new class of a-priori estimates for stationary mean-field games which have a quasi-variational structure. In particular we prove W-1,W- 2 estimates for the value function u and that the players distribution m satisfies root m is an element of W-1,W- 2. We discuss further results for power-like nonlinearities and prove higher regularity if the space dimension is 2. In particular we also obtain in this last case W-2,W-p estimates for u.