  In this paper, we investigate the existence of solution for systems of
Fokker-Planck equations coupled through a common nonlinear congestion. Two
different kinds of congestion are considered: a porous media congestion or
\textit{soft} congestion and the \textit{hard} congestion given by the
constraint $\rho_1+\rho_2 \leqslant 1$. We show that these systems can be seen
as gradient flows in a Wasserstein product space and then we obtain a
constructive method to prove the existence of solutions. Therefore it is
natural to apply it for numerical purposes and some numerical simulations are
included.
