Asympotitcs for Some Singular Monge-Ampère Equations

Given a psh function $\phi \in \mathcal{E}(\mathrm{\Omega })$ and a smooth, bounded $\theta \ge 0$, it is known that one can solve the Monge-Ampère equation $\mathrm{M}\mathrm{A}({\phi }_{\theta })={\theta }^n\mathrm{M}\mathrm{A}(\phi )$, with some form of Dirichlet boundary values, by work of Ahag–Cegrell–Czyż–Hiep. Under some natural conditions, we show that ${\phi }_{\theta }$ is comparable to $\theta \phi$ on much of $\mathrm{\Omega }$; 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.