  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 $W_p$. Our approach is by
approximation with mollified SDEs. We establish a new estimate about H{\"o}lder
continuity in time of marginal density. Then we deduce that the marginal
distributions (resp. marginal densities) of the mollified SDEs converge in
$W_p$ (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.
