On Positive Solutions to Semi-linear Conformally Invariant Equations on Locally Conformally Flat Manifolds
نویسندگان
چکیده
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 Pα were 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.
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