The argument is very similar to the one of Sadullaev ([Sa],[BL, Theorem 3.2]). By [Si], there exists an open neighborhood $W$ of $V$ in ${\mathbb{C}}^{N}$ and a holomorphic retraction $r:W\to V$. We can find an open neighborhood $U$ of $V$ so that $U\subset W$ and $\|r(z)-z\|<2$ for every $z\in U$. Indeed, if $B(p,r)$ denotes the open ball in ${\mathbb{C}}^{N}$ centered at $p$ and of radius $r$, then $U_{p}=r^{-1}(B(p,1))\cap B(p,1)$ is an open neighborhood of $p\in V$, and we let $U=\bigcup_{p\in V}U_{p}$. Since $u$ is a continuous psh exhaustion function on $V$, it follows that the function $u(r(z))$ is continuous psh on $U$ and $\lim_{z\in U,\|z\|\to+\infty}u(r(z))=+\infty$.

It is well known that there exist entire functions $f_{0},\dots,f_{N}$, so that $V=\{z\in{\mathbb{C}}^{N}:\,f_{k}(z)=0,\;0\leq k\leq N\}$ (see [Ch, p.63]). The function $\rho=\log(\sum|f_{k}|^{2})$ is psh on ${\mathbb{C}}^{N}$ and $V=\{\rho=-\infty\}$.

Let $D$ be an open set so that $V\subset D\subset\overline{D}\subset U$. Since $\rho$ is continuous on ${\mathbb{C}}^{N}\setminus V$, we can find a convex increasing function $\chi$ on $[0,+\infty)$ which verifies for every $R\geq 0$ the following two properties:

$(i)$ $\chi(R)>R-\rho(z)$ for all $z\in{\mathbb{C}}^{N}\setminus D$ with $\|z\|=R$.

$(ii)$ $\chi(R)>u(r(z))-\rho(z)$ for all $z\in\partial D$ with $\|z\|=R$.
Then

is a continuous psh exhaustion function on ${\mathbb{C}}^{N}$ and $\widetilde{u}=u$ on $V$. ∎