Large solutions to the Monge-Ampère equations with nonlinear gradient terms: Existence and boundary behavior

In this paper, we obtain conditions about the existence and boundary behavior of (strictly) convex solutions to the Monge-Ampère equations with boundary blow-updet⁡D2u(x)=b(x)f(u(x))±|∇u|q,x∈Ω,u|∂Ω=+∞, anddet⁡D2u(x)=b(x)f(u(x))(1+|∇u|q),x∈Ω,u|∂Ω=+∞, where Ω is a strictly convex, bounded smooth domain in RN with N≥2, q∈[0,N] (or q∈[0,N)), b∈C∞(Ω) which is positive in Ω, but may vanish or blow up on the boundary, f∈C[0,∞), f(0)=0, and f is strictly increasing on [0,∞) (or f∈C(R), f(s)>0,∀s∈R, and f is strictly increasing on R).