  In this paper we study the Hausdorff dimension of a elliptic measure
$\mu_{f}$ in space associated to a positive weak solution to a certain
quasilinear elliptic PDE in an open subset and vanishing on a portion of the
boundary of that open set. We show that this measure is concentrated on a set
of $\sigma-$finite $n-1$ dimensional Hausdorff measure for $p>n$ and the same
result holds for $p=n$ with an assumption on the boundary.
  We also construct an example of a domain in space for which the corresponding
measure has Hausdorff dimension $\leq n-1-\delta$ for $p\geq n$ for some
$\delta$ which depends on various constants including $p$.
  The first result generalizes the authors previous work when the PDE is the
$p-$Laplacian and the second result generalizes the well known theorem of Wolff
when $p=2$ and $n=2$.
