Refined Boundary Behavior of the Unique Convex Solution to a Singular Dirichlet Problem for the Monge–Ampère Equation

Abstract

Abstract This paper is concerned with the boundary behavior of the unique convex solution to a singular Dirichlet problem for the Monge–Ampère equation det ⁡ D 2 ⁢ u = b ⁢ ( x ) ⁢ g ⁢ ( - u ) , u < 0 , x ∈ Ω , u | ∂ ⁡ Ω = 0 , \operatorname{det}D^{2}u=b(x)g(-u),\quad u<0,\,x\in\Omega,\qquad u|_{\partial% \Omega}=0, where Ω is a strictly convex and bounded smooth domain in ℝ N {\mathbb{R}^{N}} , with N ≥ 2 {N\geq 2} , g ∈ C 1 ⁢ ( ( 0 , ∞ ) , ( 0 , ∞ ) ) {g\in C^{1}((0,\infty),(0,\infty))} is decreasing in ( 0 , ∞ ) {(0,\infty)} and satisfies lim s → 0 + ⁡ g ⁢ ( s ) = ∞ {\lim_{s\rightarrow 0^{+}}g(s)=\infty} , and b ∈ C ∞ ⁢ ( Ω ) {b\in C^{\infty}(\Omega)} is positive in Ω, but may vanish or blow up on the boundary. We find a new structure condition on g which plays a crucial role in the boundary behavior of such solution.