We study the isoperimetric problem for the axially symmetric sets in the Heisenberg
group $H^n$ with density $
^p$. At first, we prove the existence of weighted isoperimetric sets.Then,we characterize weighted isoperimetric sets uniquely as bubble sets. Finally, we deduce an interesting result that, up to a constant multiplicator, $
^p$ is the only horizontal radial density for which bubble sets can be weighted isoperimetric sets.