Existence of large singular solutions of conformal scalar curvature equations in Sn

We prove that every positive function in C1(Sn), n 6, can be approximated in the C1(Sn) norm by a positive function K ∈ C1(Sn) such that the conformal scalar curvature equation − u + n(n − 2) 4 u = Ku n+2 n−2 in Sn (0.1) has a weak positive solution u whose singular set consists of a single point. Moreover, we prove there does not exist an apriori bound on the rate at which such a solution u blows up at its singular point. Our result is in contrast to a result of Caffarelli, Gidas, and Spruck which states that Eq. (0.1), with K identically a positive constant in Sn, n 3, does not have a weak positive solution u whose singular set consists of a single point. © 2005 Elsevier Inc. All rights reserved