LAUSR

Abstract In this paper, we study the existence of positive functions K ∈ C 1 ( S n ) such that the conformal Q-curvature equation (1) P m ( v ) = K v n + 2 m n − 2 m on S n 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 ≥ 2 m + 4 , every positive function in C 1 ( S n ) can be approximated in the C 1 ( S n ) norm by a positive function K ∈ C 1 ( S n ) 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 S n , n > 2 m , does not exist a singular positive solution whose singular set is a single point.