  We study a system of partial differential equations used to describe Bertrand
and Cournot competition among a continuum of producers of an exhaustible
resource. By deriving new a priori estimates, we prove the existence of
classical solutions under general assumptions on the data. Moreover, under an
additional hypothesis we prove uniqueness.
  Keywords: mean field games, Hamilton-Jacobi, Fokker-Planck, coupled systems,
optimal control, nonlinear partial differential equations
