Electronic Journal of Differential Equations,
Vol. 2004(2004), No. 48, pp. 1-24.

Title: Dirichlet problem for degenerate
       elliptic complex Monge-Ampere equation
       
Author: Saoussen Kallel-Jallouli (Campus Univ., Tunis, Tunisie)

Abstract: 
 We consider the Dirichlet problem
 $$
 \det \big({\frac{\partial^2u}{\partial z_i\partial \overline{z_j}}}
 \big)=g(z,u)\quad\mbox{in }\Omega\,,  \quad
 u\big|_{ \partial \Omega }=\varphi\,,
 $$
 where $\Omega$ is a bounded open set of $\mathbb{C}^{n}$ with regular
 boundary, $g$ and $\varphi$ are sufficiently smooth functions, and $g$ is
 non-negative. We prove that, under additional hypotheses on $g$ and
 $\varphi $, if $|\det \varphi _{i\overline{j}}-g|_{C^{s_{\ast}}}$ is
 sufficiently small the problem has a plurisubharmonic solution.
  
Submitted May 15, 2003. Published April 6, 2004.
Math Subject Classifications: 35J70, 32W20, 32W05.
Key Words: Degenerate elliptic; omplex Monge-Ampere; Plurisubharmonic function.