Acta mathematica scientia,Series A ›› 2023, Vol. 43 ›› Issue (1): 181-202.

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On the Optimal Global Estimates of Boundary Blow-up Solutions to the Monge-Ampère Equation

Meiqiang Feng1,*(),Xuemei Zhang2()   

  1. 1School of Applied Science, Beijing Information Science & Technology University, Beijing 100192
    2School of Mathematics and Physics, North China Electric Power University, Beijing 102206
  • Received:2021-11-24 Revised:2022-04-24 Online:2023-02-26 Published:2023-03-07
  • Supported by:
    Beijing Natural Science Foundation of China(1212003)


This paper is dedicated to studying the optimal global estimates and nonexistence of strictly convex solutions to the boundary blow-up Monge-Ampère problem $ M[u](x)=K(x)f(u) \mbox{ for } x \in \Omega,\; u(x)\rightarrow +\infty \mbox{ as } {\rm dist}(x,\partial \Omega)\rightarrow 0. $ Here $M[u]=\det\, (u_{x_{i}x_{j}})$ is the Monge-Ampère operator, and $\Omega$ denotes a smooth, bounded, strictly convex domain in $ \Bbb R^N (N\geq 2)$. The interesting features in our proof are that we not only obtain the relations among various conditions imposed on $K(x)$ and $f(u)$, but make comparison of some results of global estimates in previous literatures and make clear what conditions lead to what estimations. Moreover, when $\Omega$ is a general region, we give some nonexistence results which is rarely discussed in previous literatures.

Key words: Monge-Ampère equation, Boundary blow-up, Global estimates, Strictly convex solution, Nonexistence

CLC Number: 

  • O177.91