Approximation of $m-$subharmonic functions on bounded domains in ${\mathbb{C}}^n$

Let $D$ be a bounded domain in ${\mathbb{C}}^n$. We study approximation of (not necessarily bounded from above) $m-$subharmonic function $D$ by continuous $m-$subharmonic ones defined on neighborhoods of $\bar{D}$. We also consider the existence of a $m-$subharmonic function on $D$ whose boundary values coincides with a given real valued continuous function on $\mathrm{\partial }D$ except for a sufficiently small subset of $\mathrm{\partial }D\mathrm{.}$