  We prove the existence of weak solutions to McKean-Vlasov SDEs defined on a
domain $D \subseteq \mathbb{R}^d$ with continuous and unbounded coefficients
that satisfy Lyapunov type conditions, where the Lyapunov function may depend
on measure. We propose a new type of {\em integrated} Lyapunov condition, where
the inequality is only required to hold when integrated against the measure on
which the Lyapunov function depends , and we show that this is sufficient for
the existence of weak solutions to McKean-Vlasov SDEs defined on $D$. The main
tool used in the proofs is the concept of a measure derivative due to Lions. We
prove results on uniqueness under weaker assumptions than that of global
Lipschitz continuity of the coefficients.
