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In this paper, we study weak solutions to complex Monge-Amp\`ere equations ofthe form $(\omega + dd^c \varphi)^n= F(\varphi,.)d\mu$ on a bounded strictlypseudoconvex domain in $\mathbb{C}^n$, where $\omega$ is a smooth $(1,1)$-form,$0\leq F$ is a continuous non-decreasing function, and $\mu$ is a positivenon-pluripolar measure. Our results extend previous works of Ko{\l}odziej andNguyen \cite{KN15,KN23a,KN23b} who study bounded solutions, as well as Cegrell\cite{Ceg98,Ceg04,Ceg08}, Czy\.z \cite{Cz09}, Benelkourchi \cite{Ben09,Ben15}and others who treat the case when $\omega=0$ and/or $F=1$.

Degenerate complex Monge-Amp\`ere equations with non-K\"ahler forms in bounded domains