Harnack inequalities for equations of type prescribed scalar curvature

TL;DR

We study a Yamabe-type equation $\Delta u + h u = V u^{(n+2)/(n-2)}$ on $n \ge 4$ dimensional manifolds and establish Harnack-type estimates linking local $\sup_K u$ to the global $\inf_M u$. The authors perform blow-up analysis, construct rescaled solutions that converge to a standard bubble, and apply the moving-plane method to obtain either a compactness result or explicit $(1-\epsilon)$-power bounds, with distinct forms for $n=4$ and $n \ge 5$ (requiring $\epsilon > (n-4)/(n-2)$). They further derive a local Harnack inequality around each point for subsequences, and present a compactness criterion that characterizes when the local inequality forces global compactness. Overall, the work advances a priori estimate techniques for prescribed scalar curvature equations on manifolds and clarifies the role of blow-up, rolling-up, and distortion phenomena in the $n \ge 4$ setting.

Abstract

We give Harnack inequalities for solutions of equations of type prescribed scalar curvature in dimensions n $\ge$ 4.