  In this paper we introduce the p-Laplace equations for the intermediate
Schouten curvature in conformal geometry. These p-Laplace equations provide
more tools for the study of geometry and topology of manifolds. First, the
positivity of the intermediate Schouten curvature yields the vanishing of Betti
numbers on locally conformally flat manifolds as consequences of the
B\"{o}chner formula as in the works of Nayatani and Guan-Lin-Wang. Secondly and
more interestingly, when the intermediate Schouten curvature is nonnegative,
these p-Laplace equations facilitate the geometric applications of
p-superharmonic functions and the nonlinear potential theory. This leads to the
estimates on Hausdorff dimension of singular sets and vanishing of homotopy
groups that is inspired by and extends the work of Schoen-Yau. In the
forthcoming paper we will present our results on the asymptotic behavior of
p-superharmonic functions at singularities.
