Asympotitcs for Some Singular Monge-Amp\`{e}re Equations

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.