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-up det ⁡ D 2 u ( x ) = b ( x ) f ( u ( x ) ) ± | ∇ u | q , x ∈ Ω , u | ∂ Ω = + ∞ , and det ⁡ D 2 u ( x ) = b ( x ) f ( u ( x ) ) ( 1 + | ∇ u | q ) , x ∈ Ω , u | ∂ Ω = + ∞ , where Ω is a strictly convex, bounded smooth domain in R N 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 ). [ABSTRACT FROM AUTHOR]