p-Laplace equations in conformal geometry

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 Böchner formula. Second 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 dimensions of singular sets and vanishing of homotopy groups. In the forthcoming paper (Liu et al. (2023)), we present our results on the precise asymptotic behavior of p-superharmonic functions at singularities.