Some Compactness Results Related to Scalar Curvature Deformation

Abstract

Motivated by the prescribing scalar curvature problem, we study the equation $\hat{I}{”}_{g}u+K{u}^p=0(1+\hat{I}\P \le p\le \frac{n+2}{n-2})$ on locally conformally flat manifolds $(M,g)$ with $R(g)=0$. We prove that when $K$ satisfies certain conditions and the dimension of $M$ is 3 or 4, any solution $u$ of this equation with bounded energy has uniform upper and lower bounds. Similar techniques can also be applied to prove that on 4-dimensional scalar positive manifolds the solutions of $\hat{I}{”}_{g}u-\frac{n-2}{4(n-1)}R(g)u+K{u}^p=0,K>0,1+\hat{I}\P \le p\le \frac{n+2}{n-2}$ can only have simple blow-up points.