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

2005/09/19 by Jie Qing, Qing, Jie, David Raske +1 · 4 citations
Computer Science · Mathematics · #Advanced Mathematical Modeling in Engineering #Geometric Analysis and Curvature Flows #Nonlinear Partial Differential Equations #math.AP #math.DG #msc:30F45 #msc:35J30 #msc:58J50

paper · pdf · doi:10.48550/arxiv.math/0509415

16 pages

arxiv created 2005/09/19 · arxiv updated 2009/12/01

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_α 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 \frac n-α2 for some α∈ [2, n), the set of positive smooth solutions to the equation P_αu = u^\frac n+αn-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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• Potential theory and applications in conformal geometry
• Classification of fractional, singular Yamabe metrics on a twice punctured sphere I
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• The GJMS operators in geometry, analysis, and physics

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