Well-posedness of McKean-Vlasov SDEs with density-dependent drift

In this paper, we study the well-posedness of McKean-Vlasov stochastic differential equations (SDE) whose drift depends pointwisely on marginal density and satisfies a condition about local integrability in time-space variables. The drift is assumed to be Lipschitz continuous in distribution variable with respect to Wasserstein metric Wₚ. Our approach is by approximation with mollified SDEs. We establish a new estimate about H\"older continuity in time of marginal density. Then we deduce that the marginal distributions (resp. marginal densities) of the mollified SDEs converge in Wₚ (resp. topology of compact convergence) to the solution of the Fokker-Planck equation associated with the density-dependent SDE. We prove strong existence of a solution. Weak and strong uniqueness are obtained when p=1, the drift coefficient is bounded, and the diffusion coefficient is distribution free.