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# Advanced Nonlinear Studies

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Volume 18, Issue 2

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

Zhijun Zhang
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• School of Mathematics and Information Science, Yantai University, Yantai 264005, Shandong, P. R. China
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Published Online: 2018-01-10 | DOI: https://doi.org/10.1515/ans-2017-6045

## Abstract

This paper is concerned with the boundary behavior of the unique convex solution to a singular Dirichlet problem for the Monge–Ampère equation

$\mathrm{det}{D}^{2}u=b\left(x\right)g\left(-u\right),u<0,x\in \mathrm{\Omega },{u|}_{\partial \mathrm{\Omega }}=0,$

where Ω is a strictly convex and bounded smooth domain in ${ℝ}^{N}$, with $N\ge 2$, $g\in {C}^{1}\left(\left(0,\mathrm{\infty }\right),\left(0,\mathrm{\infty }\right)\right)$ is decreasing in $\left(0,\mathrm{\infty }\right)$ and satisfies ${lim}_{s\to {0}^{+}}g\left(s\right)=\mathrm{\infty }$, and $b\in {C}^{\mathrm{\infty }}\left(\mathrm{\Omega }\right)$ 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.

MSC 2010: 35J25; 35J65; 35J67

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## About the article

Revised: 2017-12-06

Accepted: 2017-12-08

Published Online: 2018-01-10

Published in Print: 2018-04-01

Funding Source: National Natural Science Foundation of China

Award identifier / Grant number: 11571295

This work is supported in part by NSF of P. R. China under grant 11571295.

Citation Information: Advanced Nonlinear Studies, Volume 18, Issue 2, Pages 289–302, ISSN (Online) 2169-0375, ISSN (Print) 1536-1365,

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