  In this paper, we prove the existence of classical solutions for second order
stationary mean-field game systems. These arise in ergodic (mean-field) optimal
control, convex degenerate problems in calculus of variations, and in the study
of long-time behavior of time-dependent mean-field games. Our argument is based
on the interplay between the regularity of solutions of the Hamilton-Jacobi
equation in terms of the solutions of the Fokker-Planck equation and
vice-versa. Because we consider different classes of couplings, distinct
techniques are used to obtain a priori estimates for the density. In the case
of polynomial couplings, we recur to an iterative method. An integral method
builds upon the properties of the logarithmic function in the setting of
logarithmic nonlinearities. This work extends substantially previous results by
allowing for more general classes of Hamiltonians and mean-field assumptions.
