Large singular solutions for conformal Q-curvature equations on [formula omitted

Abstract

In this paper, we study the existence of positive functions K∈C1(Sn) such that the conformal Q-curvature equation(1)Pm(v)=Kvn+2mn−2monSn has a singular positive solution v whose singular set is a single point, where m is an integer satisfying 1≤m<n/2 and Pm is the intertwining operator of order 2m. More specifically, we show that when n≥2m+4, every positive function in C1(Sn) can be approximated in the C1(Sn) norm by a positive function K∈C1(Sn) such that (1) has a singular positive solution whose singular set is a single point. Moreover, such a solution can be constructed to be arbitrarily large near its singularity. This is in contrast to the well-known results of Lin [24] and Wei-Xu [36] which show that Eq. (1), with K identically a positive constant on Sn, n>2m, does not exist a singular positive solution whose singular set is a single point.