Acta mathematica scientia,Series B ›› 2024, Vol. 44 ›› Issue (5): 1965-1983.doi: 10.1007/s10473-024-0520-5

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A SINGULAR DIRICHLET PROBLEM FOR THE MONGE-AMPÈRE TYPE EQUATION*

Zhijun Zhang, Bo Zhang   

  1. School of Mathematics and Information Science, Yantai University, Yantai 264005, China
  • Received:2022-08-23 Revised:2024-05-16 Online:2024-10-25 Published:2024-10-22
  • Contact: †Zhijun Zhang, E-mail,: zhangzj@ytu.edu.cn
  • About author:Bo Zhang, E-mail,: 329175332@qq.com
  • Supported by:
    Zhijun Zhang's research was supported by Shandong Provincial NSF (ZR2022MA020).

Abstract: We consider the singular Dirichlet problem for the Monge-Ampère type equation det D2u=b(x)g(u)(1+|u|2)q/2, u<0, xΩ, u|Ω=0, where Ω is a strictly convex and bounded smooth domain in Rn, q[0,n+1), gC(0,) is positive and strictly decreasing in (0,) with lims0+g(s)=, and bC(Ω) is positive in Ω. We obtain the existence, nonexistence and global asymptotic behavior of the convex solution to such a problem for more general b and g. Our approach is based on the Karamata regular variation theory and the construction of suitable sub-and super-solutions.

Key words: Monge-Ampère equation, a singular boundary value problem, the unique convex solution, global asymptotic behavior

CLC Number: 

  • 35J60
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