  We prove sharp pointwise blow-up estimates for finite-energy sign-changing
solutions of critical equations of Schr\"odinger-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 Schr\"odinger-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.
