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

TL;DR

This paper demonstrates the existence of large singular positive solutions to conformal $Q$-curvature equations on spheres, showing that such solutions can be constructed for a broad class of functions $K$, contrasting previous non-existence results.

Contribution

The authors prove that for $n \\geq 2m+4$, any positive $C^1$ function can be approximated by functions admitting singular solutions, expanding the understanding of solution existence for these equations.

Findings

Existence of singular solutions for a broad class of functions $K$.

Construction of solutions arbitrarily large near singularity.

Contrast with previous non-existence results for constant $K$.

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 $1 \leq m < n/2$ and $P_m$ is the intertwining operator of order $2m$. More specifically, we show that when $n\geq 2m+4$, 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.