On the asymptotic symmetry of singular solutions of the scalar curvature equations
We study the asymptotic behavior of positive singular solutions of the scalar curvature equations near its singular set. When the scalar curvature function is a positive constant, it was proved by Caffarelli-Gidas-Spruck that any positive singular solution with an isolated singular point at the origin asymptotically approaches to a positive entire solution in \(\mathbb{R}^n\) with a nonremovable singular point at the origin. We extend their result to the case when nonconstant scalar curvature function \(K\) is considered. When the vanishing order of \(K\) at the origin is no less then \({n-2\over 2}\), we prove that the asymptotic result of Caffarelli-Gidas-Spruck still holds. However, if the vanishing order of the scalar curvature function \(K\) is less than \({n-2\over 2}\), we show by examples that the asymptotic result does not hold.
scientific article; zbMATH DE number 1336981