A priori estimates for finite-energy sign-changing blowing-up solutions of critical elliptic equations

Abstract

We prove sharp pointwise blow-up estimates for finite-energy sign-changing solutions of critical equations of Schrödinger-Yamabe type on a closed Riemannian manifold $(M,g)$ of dimension $n\ge 3$. This is a generalisation of the so-called $C^0$-theory for positive solutions of Schrödinger-Yamabe type equations. To deal with the sign-changing case we develop a method of proof that combines an \emph{a priori} bubble-tree analysis with a finite-dimensional reduction, and reduces the proof to obtaining sharp \emph{a priori} blow-up estimates for a linear problem.

V2 of this article. Changes with respect to V1: minor typos corrected, slight change in the title, a few references added