Weak solutions for potential mean field games of controls

We analyze a system of partial differential equations that model potential mean field game controls, briefly MFGC. Such describes the interaction infinitely many negligible players competing to optimize personal value function depends in aggregate on state and, most notably, control choice all other players. A solution corresponds Nash Equilibrium, strategy for which no one player can improve by altering only their own action. investigate second order, possibly degenerate, case with non-strictly elliptic diffusion operator and local coupling function. The main result exploits potentiality employ variational techniques provide unique weak system, additional space time regularity results under assumptions. New analytical subtleties occur obtaining priori estimates introduction an distribution as well feedback.