Monge-Ampère type equation on compact Hermitian manifolds

Given a cohomology $(1,1)$-class $\{\beta \}$ of compact Hermitian manifold $(X,\omega )$ possessing a bounded potential and fixed a model potential $\phi$, motivated by Darvas-Di Nezza-Lu and Li-Wang-Zhou’s work, we show that degenerate complex Monge-Ampère equation $(\beta +d{d}^c\phi {)}^n=e^{\lambda \phi }\mu$ has a unique solution in the relative full mass class $\mathcal{E}(X,\beta ,\phi )$, where $\mu$ is a non-pluripolar measure on $X$ and $\lambda \ge 0$ is a fixed constant. As an application, we give an explicit description of Lelong numbers of elements in $\mathcal{E}(X,\beta ,\phi )$ which generalized a theorem of Darvas-Di Nezza-Lu in the Hermitian context.