On the blow-up boundary solutions of the Monge-Ampére equation with singular weights

Haitao Yang; Yibin Chang

doi:10.3934/cpaa.2012.11.697

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2012, Volume 11, Issue 2: 697-708. Doi: 10.3934/cpaa.2012.11.697

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On the blow-up boundary solutions of the Monge -Ampére equation with singular weights

Haitao Yang^1, and
Yibin Chang^1,

1.
Department of Mathematics, Zhejiang University, Hangzhou 310027

Received: June 30, 2010

Revised: June 30, 2011

Published: February 2012

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Abstract

We consider the Monge-Ampére equations det$D^2 u = K(x) f(u)$ in $\Omega$, with $u|_{\partial\Omega}=+\infty$, where $\Omega$ is a bounded and strictly convex smooth domain in $R^N$. When $f(u) = e^u$ or $f(u)= u^p$, $p>N$, and the weight $K(x)\in C^\infty (\Omega )$ grows like a negative power of $d(x)=dist(x, \partial \Omega)$ near $\partial \Omega$, we show some results on the uniqueness, nonexistence and exact boundary blow-up rate of strictly convex solutions for this problem. Existence of such solutions will be also studied in a more general case.

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Mathematics Subject Classification: Primary: 35j05, 35j25; Secondary: 35B40.

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