The Monge–Ampère type equation in the weighted pluricomplex energy class

In the paper, we prove the existence of solutions of the complex Monge–Ampere type equation -χ(u)(ddcu)n = μ in the class $\mathcal{E}_{\chi}(\Omega)$ if there exist subsolutions in this class. As an application, we prove that the complex Monge–Ampere equation (ddcu)n = μ is solvable in the class $\mathcal{E}(\Omega)$ if there exist subsolutions locally. Moreover, by an example we show that the conditions in our above result are sharp.