Let

$$M=-\min_{\zeta\in{\mathbb{P}}^{n}}u(\zeta)\geq 0.$$

We work first in an affine chart $\{z_{j}=1\}\equiv{\mathbb{C}}^{n}$. Let $X_{j}=X\cap\{z_{j}=1\}$ and let $\rho_{j}\geq 0$ be the potential of $\omega$ in this chart with $\rho_{j}(0)=0$. Then $\varphi+\rho_{j}$ is psh on $X_{j}$ and since $u\leq 0$,

$$\varphi+\rho_{j}+M<u+\rho_{j}+M\leq\frac{1}{1+\varepsilon}\,u+\rho_{j}+M\;{\rm on}\;X_{j}.$$

Note that $(1+\varepsilon)^{-1}u+\rho_{j}+M\geq 0$ is a continuous psh exhaustion function on ${\mathbb{C}}^{n}$. Theorem A yields a psh function $\widetilde{\psi}$ on ${\mathbb{C}}^{n}$ so that

$$\widetilde{\psi}<\frac{c}{1+\varepsilon}\,u+c\rho_{j}+cM\;{\rm on}\;{\mathbb{C}}^{n}\;,\;\;\widetilde{\psi}=\varphi+\rho_{j}+M\;{\rm on}\;X_{j}.$$

The function $\psi_{j}=\widetilde{\psi}-c\rho_{j}-cM$ extends uniquely to a $c\omega$-psh function on ${\mathbb{P}}^{n}$ which verifies

$$\psi_{j}\leq\frac{c}{1+\varepsilon}\,u\;\;{\rm on}\;{\mathbb{P}}^{n}.$$

Moreover on $X\cap\{z_{j}=1\}$ we have

$$\psi_{j}=\varphi-(c-1)\rho_{j}-(c-1)M=\varphi+(c-1)\theta_{j}-(c-1)M,$$

where

$$\theta_{j}(z)=\log\frac{|z_{j}|}{\sqrt{|z_{0}|^{2}+\ldots+|z_{n}|^{2}}}\;.$$

Hence $\psi_{j}=-\infty$ on $X\cap\{z_{j}=0\}$.

We finally let $\psi=\max\{\psi_{0},\ldots,\psi_{n}\}$. This is a $c\omega$-psh function on ${\mathbb{P}}^{n}$ which verifies the desired conclusions, since $\theta=\max\{\theta_{0},\ldots,\theta_{n}\}$. ∎