Dirichlet problem for complex Monge-Ampere equation near an isolated KLT singularity

We solve the Dirichlet problem for complex Monge-Ampere equation near an isolate Klt singularity, which generalizes the result of Eyssidieux et al. [Singular Kahler-Einstein metrics, J. Amer. Math. Soc. 22 (2009) 607-639], where the Monge-Ampere equation is solved on singular varieties without boundary. As a corollary, we construct solutions to Monge-Ampere equation with isolated singularity on strongly pseudoconvex domain Omega contained in C-n.