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In this paper, we study the existence of positive functions $K \inC^1(\mathbb{S}^n)$ such that the conformal $Q$-curvature equation\begin{equation}\label{001} P_m (v) =K v^{\frac{n+2m}{n-2m}} {on} ~\mathbb{S}^n \{equation} has a singular positive solution $v$ whose singularset is a single point, where $m$ is an integer satisfying $1 \leq m < n/2$ and$P_m$ is the intertwining operator of order $2m$. More specifically, we showthat when $n\geq 2m+4$, every positive function in $C^1(\mathbb{S}^n)$ can beapproximated in the $C^1(\mathbb{S}^n)$ norm by a positive function $K\inC^1(\mathbb{S}^n)$ such that the conformal $Q$-curvature equation has asingular positive solution whose singular set is a single point. Moreover, sucha solution can be constructed to be arbitrarily large near its singularity.This is in contrast to the well-known results of Lin \cite{Lin1998} and Wei-Xu\cite{Wei1999} which show that the conformal $Q$-curvature equation, with $K$identically a positive constant on $\mathbb{S}^n$, $n > 2m$, does not exist asingular positive solution whose singular set is a single point.

Large singular solutions for conformal $Q$-curvature equations on $\mathbb{S}^n$