Classical solutions to local first-order extended mean field games

We study the existence of classical solutions to a broad class local, first order, forward-backward extended mean field games systems, that includes standard games, with congestion, and type control problems. work strictly monotone cost may be fully coupled Hamiltonian, which is assumed have superlinear growth. Following previous on order system, we prove smooth under coercivity condition ensures positive density players, assuming strict form uniqueness for system. Our relies transforming problem into partial differential equation oblique boundary conditions, elliptic precisely condition.