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

AbstractWe 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(0.1)-Δu+n(n-2)4u=Kun+2n-2inSnhas 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