Degenerate complex Monge-Ampère equations with non-Kähler forms in bounded domains

In this paper, we study weak solutions to complex Monge-Ampère equations of the form $(\omega +d{d}^c\phi {)}^n=F(\phi ,\mathrm{.})d\mu$ on a bounded strictly pseudoconvex domain in ${\mathbb{C}}^n$, where $\omega$ is a smooth $(1,1)$-form, $0\le F$ is a continuous non-decreasing function, and $\mu$ is a positive non-pluripolar measure. Our results extend previous works of Kołodziej and Nguyen \cite{KN15,KN23a,KN23b} who study bounded solutions, as well as Cegrell \cite{Ceg98,Ceg04,Ceg08}, Czyż \cite{Cz09}, Benelkourchi \cite{Ben09,Ben15} and others who treat the case when $\omega =0$ and/or $F=1$.