Some Compactness Results Related to Scalar Curvature Deformation

Abstract

Motivated by the prescribing scalar curvature problem, we study the equation $\Delta_g u +Ku^p=0 (1+\zeta \leq p \leq \frac{n+2}{n-2})$ 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 $\Delta_gu-\frac{n-2}{4(n-1)}R(g)u+Ku^p=0, K>0, 1+\zeta \leq p \leq \frac{n+2}{n-2}$ can only have simple blow-up points.