In the talk we first give a brief survey on complex Monge-Ampère equations
with special boundary data or prescribed singularities that naturally occur in analysis and
geometry. We then discuss a theorem due to work Cheng and Yau on the existence of
complete Kähler-Einstein metrics on pseudoconvex domains in C^n. We seek a more direct
approach to the problem from PDE point of view, and present a method with potential of broader applications.