A priori estimates for prescribing scalar curvature equations

Abstract

We obtain a priori estimates for solutions to the prescribing scalar curvature equation (1) - △u + n(n - 2)/4u = n - 2/4(n-1)R(x)un+2/n-2 on Sn for n ≥ 3. There have been a series of results in this respect. To obtain a priori estimates people required that the function R(x) be positive and bounded away from 0. This technical assumption has been used by many authors for quite a few years. It is due to the fact that the standard blowing-up analysis fails near R(x) = 0. The main objective of this paper is to remove this well-known assumption. Using the method of moving planes, we are able to control the growth of the solutions in the region where R is negative and in the region where R is small, and thus obtain a priori estimates on the solutions of (1) for a general function R which is allowed to change signs.

Department(s)

Mathematics

Document Type

Article

DOI

https://doi.org/10.2307/2951844

Publication Date

1-1-1997

Journal Title

Annals of Mathematics

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