On positive solutions to semi-linear conformally invariant equations on locally conformally flat manifolds

In this paper we study the existence and compactness of positive solutions to a family of conformally invariant equations on closed locally conformally flat manifolds. The family of conformally covariant operators $P_{\alpha }$ were introduced via the scattering theory for Poincar'{e} metrics associated with a conformal manifold $(M^n,[g])$. We prove that, on a closed and locally conformally flat manifold with Poincar'{e} exponent less than $\frac {n-\alpha}2$forsome$\alpha \in [2, n)$, the set of positive smooth solutions to the equation $$P_{\alpha }u=u^{\frac{n+\alpha }{n-\alpha }}$$ is compact in the $C^{\mathrm{\infty }}$ topology. Therefore the existence of positive solutions follows from the existence of Yamabe metrics and a degree theory.