Date: 22 Apr 2006

Asymptotic symmetries for conformal scalar curvature equations with singularity

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Abstract

We give conditions on a positive Hölder continuous function C2such that every C 2 positive solution u((x)) of the conformal scalar curvature equation

\(\Delta u+K(x)u^{\frac{n+2}{n-2}}=0 \)

in a punctured neighborhood of the origin in R n either has a removable singularity at the origin or satisfies

\(u(x)=u_0(|x|)(1+ {\cal O} (|x|^\beta)) \quad \text{as} \quad |x|\to 0^+\)

for some positive singular solution u 0(|x|) of

\(\Delta u_0+K(0)u_0^{\frac{n+2}{n-2}}=0 \quad \text{in}\quad {\bf R}^n\setminus \{0\}\)

where \(\beta\in(0,1)\) is the Hölder exponent of K.Mathematics Subject Classification (2000) Primary 35J60, 53C21