Abstract

We study the equation Δ_gu − (n − 2) ∕ (4(n − 1))R(g)u + Ku^p = 0 for p in 1 + ζ ≤ p ≤ (n + 2) ∕ (n − 2) on locally conformally flat compact manifolds (Mⁿ,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.

Keywords

scalar curvature, conformal deformation, uniform estimates

Mathematical Subject Classification

Primary: 53C21

Authors