Let $\mu$ and $\nu$ be measures on Radon spaces $X$ and $Y$, respectively. The Monge-Kantorovich problem is stated as $$ \inf_{\gamma \in \Gamma(\mu,\nu)} \int_{X\times Y} c(x,y) d\gamma(x,y)$$ where $\Gamma(\mu,\nu)$ is the set of joint distributions with $\mu$ and $\nu$ as marginals.
It seems there has been extensive work on existence and uniqueness in various settings (e.g. if $X=Y=\mathbb{R}^n$, if the cost is of the form $c(x,y) = h(x-y)$, etc.). Is there a general result stating requirements for existence and uniqueness of the minimizer? E.g. I think if $c$ is lower semicontinuous then there is a minimizer, but I am not sure.