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

In the paper, we prove the existence of solutions of the complex Monge–Ampère type equation −χ(u)(ddcu)n = µ in the class Eχ(Ω) if there exist subsolutions in this class. As an application, we prove that the complex Monge–Ampère equation (ddcu)n = µ is solvable in the class E(Ω) if there exist subsolutions locally. Moreover, by an example we show that the conditions in our above result are sharp.