  In \cite{CHMY04}, we studied $p$-mean curvature and the associated
$p$-minimal surfaces in the Heisenberg group from the viewpoint of PDE and
differential geometry. In this paper, we look into the problem through the
variational formulation. We study a generalized $p$-area and associated ($p$-)
minimizers in general dimensions.
  We prove the existence and investigate the uniqueness of minimizers. Since
this is reduced to solving a degenerate elliptic equation, we need to consider
the effect of the singular set and this requires a careful study. We define the
notion of weak solution and prove that in a certain Sobolev space, a weak
solution is a minimizer and vice versa. We also give many interesting examples
in dimension 2. An intriguing point is that, in dimension 2, a $C^2$-smooth
solution from the PDE viewpoint may not be a minimizer. However, this statement
is true for higher dimensions due to the relative smallness of the size of the
singular set.
