  Existence and uniqueness of a weak solution for first order mean field game
systems with local coupling are obtained by variational methods. This solution
can be used to devise $\epsilon-$Nash equilibria for deterministic differential
games with a finite (but large) number of players. For smooth data, the first
component of the weak solution of the MFG system is proved to satisfy (in a
viscosity sense) a time-space degenerate elliptic differential equation.
