  In this paper, we prove the existence and uniqueness of solutions as well as
ergodicity for McKean-Vlasov SDEs under Lyapunov conditions, in which the
Lyapunov functions are defined on $\mathbb R^d\times \mathcal P_2(\mathbb
R^d)$, i.e. the Lyapunov functions depend not only on space variable but also
on distribution variable. It is reasonable and natural to consider
distribution-dependent Lyapunov functions since the coefficients depends on
distribution variable. We apply the martingale representation theorem and a
modified Yamada-Watanabe theorem to obtain the existence and uniqueness of
solutions. Furthermore, the Krylov-Bogolioubov theorem is used to get
ergodicity since it is valid by linearity of the corresponding Fokker-Planck
equations on $\mathbb R^d\times \mathcal P_2(\mathbb R^d)$. In particular, if
the Lyapunov function depends only on space variable, we obtain exponential
ergodicity for semigroup $P_t^*$ under Wasserstein quasi-distance. Finally, we
give some examples to illustrate our theoretical results.
