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In this paper, we consider a class of infinitely degenerate partialdifferential systems to obtain the Nash equilibria in the mean field games. Thedegeneracy in the diffusion and the Hamiltonian may be different. This featurebrings difficulties to the uniform boundness of the solutions, which is centralto the existence and regularity results. First, from the perspective of thevalue function in the stochastic optimal control problems, we prove theLipschitz continuity and the semiconcavity for the solutions of theHamilton-Jacobi equations (HJE). Then the existence of the weak solutions forthe degenerate systems is obtained via a vanishing viscosity method.Furthermore, by constructing an auxiliary function, we conclude the regularityof the viscosity solution for the HJE in the almost everywhere sense.

Mean Field Games with infinitely degenerate diffusion and non-coercive Hamiltonian