Large singular solutions for conformal Q-curvature equations on Sⁿ

In this paper, we study the existence of positive functions K∈C¹(Sⁿ) such that the conformal Q-curvature equation [Formula presented] has a singular positive solution v whose singular set is a single point, where m is an integer satisfying 1≤m<n/2 and P_m 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.