This paper develops the existence and uniqueness of solutions to McKean-Vlasov stochastic differential equations (SDEs). One of the main novelties is the use of one-sided local Lipschitz condition on the drift and local Lipschitz condition on the diffusion coefficient both with respect to the state variable. For any neighborhood with radius $R$, the local Lipschitz constants of the drift and diffusion coefficients are of the orders $O(\log R)$ and $O(\sqrt{\log R})$, respectively. Owing to the distribution-dependent coefficients, standard techniques developed for classical SDEs are no longer applicable. New techniques including Euler-like approximation are developed to overcome the difficulties. Moreover, this paper establishes a sufficient condition for the $p$th moment exponential stability of McKean-Vlasov SDEs by using measure-dependent Lyapunov functions.

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