We consider the singular Dirichlet problem for the Monge-Ampère type equation ${\rm det} \ D^2 u=b(x)g(-u)(1+|\nabla u|^2)^{q/2}, \ u<0, \ x \in \Omega, \ u|_{\partial \Omega}=0,$ where $\Omega$ is a strictly convex and bounded smooth domain in $\mathbb R^n$, $q\in [0, n+1)$, $g\in C^\infty(0,\infty)$ is positive and strictly decreasing in $(0, \infty)$ with $\lim\limits_{s \rightarrow 0^+}g(s)=\infty$, and $b \in C^{\infty}(\Omega)$ is positive in $\Omega$. 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.
Zhijun Zhang
,
Bo Zhang
. A SINGULAR DIRICHLET PROBLEM FOR THE MONGE-AMPÈRE TYPE EQUATION*[J]. Acta mathematica scientia, Series B, 2024
, 44(5)
: 1965
-1983
.
DOI: 10.1007/s10473-024-0520-5
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