Asymptotic symmetries for conformal scalar curvature equations with singularity

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