Nonlinear elliptic systems and mean-field games



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 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. 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.

Mathematics Subject Classification

35Q91 49N70 91A10 


Nonlinear elliptic systems Stochastic differential games N-person games Nash equilibria Mean-Field Games 


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Copyright information

© Springer International Publishing 2016

Authors and Affiliations

  1. 1.Dipartimento di MatematicaUniversità di PadovaPaduaItaly
  2. 2.Mathematics InstituteCardiff UniversityCardiffUK

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