On positive solutions to semilinear conformally invariant equations on locally conformally flat manifolds

Abstract

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 Pα was introduced via the scattering theory for Poincaré metrics associated with a conformal manifold (Mn,[g]). We prove that, on a closed and locally conformally flat manifold with Poincaré exponent less than (n−α)/2 for some α∈[2,n), the set of positive smooth solutions to the equation Pαu=u(n+α)/(n−α) is compact in the C∞ topology. Therefore the existence of positive solutions follows from the existence of Yamabe metrics and a degree theory.