A compactness theorem on Branson's $Q$-curvature equation

Abstract: Let $(M, g)$ be a closed Riemannian manifold of dimension $5$. Assume that $(M, g)$ is not conformally equivalent to the round sphere. If the scalar curvature $R_g\geq 0$ and the $Q$-curvature $Q_g\geq 0$ on $M$ with $Q_g(p)>0$ for some point $p\in M$, we prove that the set of metrics in the conformal class of $g$ with prescribed constant positive $Q$-curvature is compact in $C^{4, \alpha}$ for any $0 <\alpha < 1$. We also give some estimates for dimension $6$ and $7$.