The scalar curvature deformation equation on locally conformally flat manifolds of higher dimensions

We study the equation Delta(g)u -(n - 2)/(4(n - 1)) R(g) u + Ku(p) = 0 for p in 1 + zeta <= p <= (n + 2)/(n - 2) on locally conformally flat compact manifolds (M-n, g). We prove that when the scalar curvature R(g) 0 and n >= 5, under suitable conditions on K, all positive solutions u with bounded energy have uniform upper and lower bounds. In our previous 2007 paper, we also assumed this energy bound condition for the uniform estimates in the lower-dimensional case. We now give an example showing that this condition is necessary.