Inserted: 19 mar 2015
Last Updated: 9 mar 2017
Journal: NoDEA Nonlinear Differential Equations Appl.
We consider a class of quasilinear elliptic systems of PDEs consisting of N Hamilton-Jacobi-Bellman equations coupled with N divergence form equations, generalising to N>1 populations the PDEs for stationary Mean-Field Games first proposed by Lasry and Lions. We provide two alternative sets of sufficient conditions for the existence of solutions to these systems. The first requires that the Hamiltonians grow faster than linearly in the gradient variables and do not oscillate too much in the space variables, similarly to Lasry and Lions. The second prescribes instead that the Hamiltonians behave at most linearly for large gradients, therefore modelling games where the controls of the agents are bounded; this is new in the current context. We show the connection of these systems with the classical strongly coupled systems of Hamilton-Jacobi-Bellman equations of the theory of N-person stochastic differential games studied by Bensoussan and Frehse. We also prove the existence of Nash equilibria in feedback form for some N-person games.
Keywords: Elliptic systems, Hamilton-Jacobi-Bellman equations, Mean-Field Games, N-person stochastic differential games