  Let $1\leq m\leq n$ be two fixed integers. Let $\Omega \Subset \mathbb C^n$
be a bounded $m$-hyperconvex domain and $\mathcal A \subset \Omega \times ]0,+
\infty[$ a finite set of weighted poles. We define and study properties of the
$m$-subharmonic Green function of $\Omega$ with prescribed behaviour near the
weighted set $A$. In particular we prove uniform continuity of the exponential
Green function in both variables $(z,\mathcal A)$ in the metric space $\bar
\Omega \times \mathcal F$, where $\mathcal F$ is a suitable family of sets of
weighted poles in $\Omega \times ]0,+ \infty[$ endowed with the Hausdorff
distance. Moreover we give a precise estimates on its modulus of continuity.
Our results generalize and improve previous results concerning the pluricomplex
Green function du to P. Lelong.
