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

By J. Qing, D. Raske. 2005

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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.

Subfields: Applied Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/1a514c38cdbdc67add6d4e06f39c05d723af49ab

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In Applied Mathematics

In Geometry and Topology

Citations in the corpus

On a fractional Nirenberg problem, part I: blow up analysis and compactness of solutions (2011, background): “Fractional Yamabe problems about finding constant σ -curvatures on the conformal infinity of given asymptotically hyperbolic manifolds have been studied by González, Mazzeo and Sire [54], González and Qing [55] and Qing and Raske [87].”
Fractional Laplacian in conformal geometry (2010, background): “On one hand, there are extensive works that study properties of fractional Laplacian operators as non-local operators together with applications to free-boundary value problems and non-local minimal surfaces - by CaffarelliSilvestre [6], and many others (see the related articles [5], [7], [4], [9]);”
The Nirenberg problem and its generalizations: a unified approach (2014, background): “Other work which are closely related to these fractional Q-curvatures includes: prescribing fractional order curvature problems on general manifolds by Qing and Raske [67], González et al.”
Existence theorems of the fractional Yamabe problem (2016, background): “2) was obtained in [39] for a locally con formally flat manifold (n ≥ 5) with positive Yamabe constant and Poincaré exponent less than ( n− 4)/2.”
Fractional conformal Laplacians and fractional Yamabe problems (2010, background): “The conformal fractional Laplacians and fractional scalar curvatures should also be compared to the higher order generalization of the conformal Laplacian and scalar curvature: the Paneitz operator P ĥ 2 and its associated Q-curvature (see [38], [5], [39]).”