. Let $X$ be a compact Kähler manifold of complex dimension $n$, let $\Omega>0$ be a smooth volume form, let $\theta$ be a continuous closed positive $(1,1)$-form such that $\{\theta^{n}=0\}$ is a set of measure $0$, let $\gamma$ be a closed positive $(1,1)$-current with continuous local potentials such that $\gamma\geq\eta\,\theta$, $\eta>0$. Let also $f\in L\log^{n+\delta}L(X)$, $\delta>0$ such that $\int_{X}\gamma^{n}=\int_{X}f\,\Omega$ and $\lambda\geq 0$ be a real number. Then there exists a unique solution $\psi\in\hat{\cal P}_{\gamma}$ of the degenerate complex Monge-Ampère equation

$$(\gamma+i\partial\bar{\partial}\psi)^{n}=f\,e^{\lambda\,\psi}\Omega\,,$$

which in the case $\lambda=0$ is normalized by $0=\max_{X}\psi$. The solution is continuous and satisfies the $C^{0}$-estimate $\|\psi\|_{C^{0}(X)}\leq C(\delta,\gamma,\Omega)\,I_{\delta}(f)^{\frac{n}{\delta}}+1$, with

$$I_{\delta}(f):=\{\gamma\}^{-n}\int\limits_{X}f\log^{n+\delta}\left(e+\{\gamma\}^{-n}f\right)\Omega\,.$$

Moreover the constant $C(\delta,\gamma,\Omega)>0$ stays bounded for perturbations of $\gamma\geq 0$ as in the statement (C) of theorem 3.