The Scalar Curvature Deformation Equation on Locally Conformally Flat Manifolds

Abstract

We study the equation $\Delta_g u -\frac{n-2}{4(n-1)}R(g)u+Ku^p=0 (1+\zeta \leq p \leq \frac{n+2}{n-2})$ on locally conformally flat compact manifolds $(M^n,g)$. We prove the following: (i) When the scalar curvature $R(g)>0$ and the dimension $n \geq 4$, 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 $n \geq 5$, 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.