On a general class of free boundary Monge-Ampère equations

We solve a general class of free boundary Monge-Ampère equations given by $$\det D^{2}u=\lambda {\displaystyle \frac{f(-u)}{g(u^{\star })h(\mathrm{\nabla }u)}}{\chi }_{\{u<0\}}\text{ in }{\mathbb{R}}^n,\mathrm{\nabla }u({\mathbb{R}}^n)=P$$ where $P$ is a bounded convex set containing the origin, and $h>0$ on $P$. We consider applications to optimal transport with degenerate densities, Monge-Ampère eigenvalue problems, and geometric problems including a hemispherical Minkowski problem and free boundary Kähler-Ricci solitons on toric Fano manifolds.