Boundary Behavior of Large Solutions for Equations of Monge-Ampère Type

Abstract

This paper is concerned with the existence and optimal boundary behavior of large solutions to the Monge-Ampère type equations det D²u(x) = λuⁿ(x)+b(x)g(∣∇u(x)∣), x ∈ Ω, where Ω is a uniformly convex, bounded smooth domain in ℝⁿ with n ≥ 2, b ∈ C^∞(Ω) is positive in Ω, g ∈ C[0, ∞) ∩ C¹ (0, ∞), g(0) = 0 and g is increasing on [0, ∞). The author finds new structure conditions on g which play a crucial role in boundary behavior of such solutions.

Keywords

Monge-Ampère type equations / A nonlinear gradient term / Large convex solutions / Boundary behavior / 35J25 / 35J92

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