A priori estimates for prescribing scalar curvature equations
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.
Chen, Wenxiong, and Congming Li. "A priori estimates for prescribing scalar curvature equations." Annals of mathematics 145, no. 3 (1997): 547-564.
Annals of Mathematics