The Scalar Curvature Deformation Equation on Locally Conformally Flat Manifolds

Abstract

We study the equation $\hat{I}{”}_{g}u-\frac{n-2}{4(n-1)}R(g)u+K{u}^p=0(1+\hat{I}\P \le p\le \frac{n+2}{n-2})$ on locally conformally flat compact manifolds $(M^n,g)$. We prove the following: (i) When the scalar curvature $R(g)>0$ and the dimension $n\ge 4$, under suitable conditions on $K$, all positive solutions $u$ have uniform upper and lower bounds; (ii) When the scalar curvature $R(g)\equiv 0$ and $n\ge 5$, under suitable conditions on $K$, all positive solutions $u$ with bounded energy have uniform upper and lower bounds. We also give an example to show that the energy bound condition for the uniform estimates in math.DG/0602636 is necessary.