A priori estimates for prescribing scalar curvature equations


We obtain a priori estimates for solutions to the prescribing scalar curvature equation
-\Delta u + \frac{n(n-2)}{4} u = \frac{n-2}{4(n-1)} R(x) u^{\frac{n+2}{n-2}}
on $S^n$ or $n\ge 3$. 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 techncial 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.




Wenxiong Chen

Congming Li