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

Abstract

In this paper, we study the existence of positive functions $K\in C^1({\mathbb{S}}^n)$ 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 singular set is a single point, where $m$ is an integer satisfying $1\le m<n/2$ and $P_m$ is the intertwining operator of order $2m$. More specifically, we show that when $n\ge 2m+4$, every positive function in $C^1({\mathbb{S}}^n)$ can be approximated in the $C^1({\mathbb{S}}^n)$ norm by a positive function $K\in C^1({\mathbb{S}}^n)$ such that the conformal $Q$-curvature equation 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 \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 a singular positive solution whose singular set is a single point.

arXiv comment24 pages; fixed some typos

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