Asympotitcs for Some Singular Monge-Ampère Equations

Abstract: Given a psh function $\varphi\in\mathcal{E}(\Omega)$ and a smooth, bounded $\theta\geq 0$, it is known that one can solve the Monge-Amp`{e}re equation $\mathrm{MA}(\varphi_\theta)=\theta^{n\mathrm{MA}(\varphi)$,} with some form of Dirichlet boundary values, by work of Ahag--Cegrell--Czy.{z}--Hiep. Under some natural conditions, we show that $\varphi_\theta$ is comparable to $\theta\varphi$ on much of $\Omega$; especially, it is bounded on the interior of ${\theta = 0}$. Our results also apply to complex Hessian equations, and can be used to produce interesting Green's functions.