I will talk about a class of quasilinear elliptic systems of PDEs consisting of $N$ Hamilton-Jacobi-Bellman equations coupled with $N$ Fokker-Planck equations, generalising to $N>1$ populations the PDEs for stationary Mean-Field Games first proposed by Lasry and Lions. I will describe a wide range of sufficient conditions for the existence of solutions to these systems: either the Hamiltonians are required to behave at most linearly for large gradients, as it occurs when the controls of the agents are bounded, or they must grow faster than linearly and not oscillate too much in the space variables, in a suitable sense. I will show the connection
of these systems with Mean Field Game problems and a class of $N$-player games.
Time permitting, I may talk also about the extension of some of these results to a class of degenerate elliptic systems with second-order operators of Hörmander type, arising from differential games where the dynamic is described by a stochastic differential equation with diffusion matrix having as columns vector fields satisfying Hörmander condition.