  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 $\frac
{n-\alpha}2$ for some $\alpha \in [2, n)$, the set of positive smooth solutions
to the equation $$ P_\alpha u = u^\frac {n+\alpha}{n-\alpha} $$ is compact in
the $C^\infty$ topology. Therefore the existence of positive solutions follows
from the existence of Yamabe metrics and a degree theory.
