Refined Asymptotic Estimates for Conformal Scalar Curvature Equation via Moving Sphere Method

In this paper, we use the so-called moving sphere method to give local estimates of a positive singular solution u near its singular set Z of the conformal scalar curvature equation Δu+K(x)u(n+2)/(n−2)=0inΩ\Z,u>0 and u∈C2in Ω\Z , where Ω⊃ B2 is an open bounded subset of Rn, n⩾3,K(x) is a continuous function defined on Ω and Z is a compact subset of Ω with Newtonian capacity zero. Under some flatness assumptions on K we show that u(x)d(x,Z)(n−2)/2⩽C.