On the solution of Dirichlet's problem of the complex Monge-Ampère equation for the Cartan-Hartogs domain of the first type

The complex Monge-Ampère equation is a nonlinear equation with high degree; therefore getting its solution is very difficult. In the present paper how to get the solution of Dirichlet's problem of the complex Monge-Ampère equation on the Cartan-Hartogs domain of the first type is discussed, using an analytic method. Firstly, the complex Monge-Ampère equation is reduced to a nonlinear ordinary differential equation, then the solution of Dirichlet's problem of the complex Monge-Ampère equation is reduced to the solution of a two-point boundary value problem for a nonlinear second-order ordinary differential equation. Secondly, the solution of Dirichlet's problem is given as a semi-explicit formula, and in a special case the exact solution is obtained. These results may be helpful for a numerical method approach to Dirichlet's problem of the complex Monge-Ampère equation on the Cartan-Hartogs domain of the first type.