Some Compactness Results Related to Scalar Curvature Deformation

Motivated by the prescribing scalar curvature problem, we study the equation ∆gu+Ku p = 0 (1 + ζ ≤ p ≤ 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 ∆gu− n−2 4(n−1)R(g)u+Ku p = 0,K > 0, 1+ζ ≤ p ≤ n+2 n−2 can only have simple blow-up points.