Optimal global asymptotic behavior of the solution to a singular monge-ampère equation

Zhijun Zhang

doi:10.3934/cpaa.2020053

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2020, Volume 19, Issue 2: 1129-1145. Doi: 10.3934/cpaa.2020053

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Optimal global asymptotic behavior of the solution to a singular monge-ampère equation

Zhijun Zhang

School of Mathematics and Information Science, Yantai University, Yantai 264005, Shandong, China

Received: March 31, 2019

Revised: May 31, 2019

Published: October 2019

The author is supported by NSF grant of P. R. China under grant 11571295

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Abstract

This paper is mainly concerned with the optimal global asymptotic behavior of the unique convex solution to a singular Dirichlet problem for the Monge-Ampère equation $ {\rm det} \ D^2 u = b(x)g(-u), \ u<0, \ x \in \Omega, \ u|_{\partial \Omega} = 0, $ where $ \Omega $ is a strict convex and bounded smooth domain in $ \mathbb R^n $ with $ n\geq 2 $, $ g\in C^1((0,\infty)) $ is positive and decreasing in $ (0, \infty) $ with $ \lim_{s \rightarrow 0^+}g(s) = \infty $, $ b \in C^{\infty}(\Omega) $ is positive in $ \Omega $, but may vanish or blow up on the boundary properly. Our approach is based on the construction of suitable sub- and super-solutions.

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Mathematics Subject Classification: Primary: 35J75; Secondary: 35J96.

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