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# A Priori Estimates for Prescribing Scalar Curvature Equations

Wenxiong Chen and Congming Li
Annals of Mathematics
Second Series, Vol. 145, No. 3 (May, 1997), pp. 547-564
DOI: 10.2307/2951844
Stable URL: http://www.jstor.org/stable/2951844
Page Count: 18
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## Abstract

We obtain a priori estimates for solutions to the prescribing scalar curvature equation \begin{equation*}\tag(1)}-\Delta u + \frac{n(n - 2)}{4} u = \frac{n - 2}{4(n - 1)} R(x)u^{\frac{n + 2}{n - 2}}\end{equation*} 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.

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