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

Abstract

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 Poincaré metrics associated with a conformal manifold $(M^n,[g])$. We prove that, on a closed and locally conformally flat manifold with Poincaré 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.

16 pages