Compactness for conformal metrics with Constant $Q$ curvature on locally conformally flat manifolds

In this note we study the conformal metrics of constant $Q$ curvature on closed locally conformally flat manifolds. We prove that for a closed locally conformally flat manifold of dimension $n\ge 5$ and with Poincar"{e} exponent less than $\frac{n-4}{2}$, the set of conformal metrics of positive constant $Q$ and positive scalar curvature is compact in the $C^{\mathrm{\infty }}$ topology.