Estimates of the conformal scalar curvature equation via the method of moving planes

✍ Scribed by Chiun-Chuan Chen; Chang-Shou Lin

• Publisher: John Wiley and Sons
• Year: 1997
• Tongue: English
• Weight: 413 KB
• Volume: 50
• Category: Article
• ISSN: 0010-3640
• DOI: 10.1002/(sici)1097-0312(199710)50:10<971::aid-cpa2>3.0.co;2-d

✦ Synopsis

In this paper we derive a local estimate of a positive singular solution u near its singular set Z of the conformal scalar curvature equation

where K(x) is a positive continuous function, Z is a compact subset of Ω, and g satisfies that

Assuming that the order of flatness at critical points of K on Z is no less than n-2 2 , we prove that, through the application of the method of moving planes, the inequality

holds for any solution of (0.1) with Cap(Z) = 0.

By the same method, we also derive a Harnack-type inequality for smooth positive solutions. Let u satisfy

Assume that the order of flatness at critical points of K is no less than n-2; then the inequality

We also show by examples that the assumption about the flatness at critical points is optimal for validity of the inequality (0.4).