Large Solutions to Complex Monge-Amp`ere Equations:Existence, Uniqueness and Asymptotics

The authors consider the complex Monge-Ampère equation det( u_i`(j)] )\left( {u_{i\bar j} } \right) = ψ(z, u, ∇ u) in bounded strictly pseudoconvex domains Ω, subject to the singular boundary condition u = ∞ on ∂Ω. Under suitable conditions on ψ, the existence, uniqueness and the exact asymptotic behavior of solutions to boundary blow-up problems for the complex Monge-Ampère equations are established.