  The existence of stationary distributions to distribution dependent
stochastic differential equations are investigated by using the ergodicity of
the associated decoupled equation and the Schauder fixed point theorem. By
using Zvonkin's transformation, we also establish the existence result for
equations with singular coefficients. Instead of the uniqueness, the
non-uniqueness of stationary distributions are considered for equations with
regular coefficients. Concrete examples including McKean-Vlasov stochastic
equations with the quadratic interaction and the non-quadratic interaction, and
equations with a bounded and discontinuous drift are presented to illustrate
our non-uniqueness results.
