Blow-up phenomena for the constant Q/R-curvature equation

Abstract

Let $n\ge 25$ be an integer. In this paper, we construct a Riemannian metric $g_{0}$ on $\mathbb{S}^n$, smooth for $n\geq 26$ and of class $C^9$ but not $C^{10}$ for $n=25$, with the property that the set of metrics in the conformal class of $g_{0}$ having positive scalar curvature and positive constant quotient $Q/R$ is non-compact. Equivalently, we construct families of solutions exhibiting blow-up behavior for the following equation \begin{align*} P _{g_{0}}u- \frac{ (n+2 )(n-4 )}{4} u^{ \frac{2}{n-4}} L_{g_{0}}u^{ \frac{n-2}{n-4}} =0, \quad u>0\quad\text{on} \ \mathbb{S}^{n}, \end{align*} where $P _{g_{0}}$ is the Paneitz operator and $ L_{g_{0}}=-\Delta_{g_{0}} +\frac{n-2}{4(n-1 )}R_{g_{0}} $ is the conformal Laplacian of $ g_{0}$.