论文标题
$ \ mathbb {s}^n $上的共形$ q $ curvature方程的大型单数解决方案
Large singular solutions for conformal $Q$-curvature equations on $\mathbb{S}^n$
论文作者
论文摘要
在本文中，我们研究了c^1（\ mathbb {s}^n）$中的积极功能的存在$ k \，以使综合$ q $ -c $ -curvature方程\ begin \ begin {qore} \ label {001} p_m（v）p_m（v）= k v^k v^k v^{ \ Mathbb {s}^n \ {equation}具有一个单一的正面解决方案$ V $，其单数是一个点，其中$ m $是满足$ 1 \ leq m <n/2 $和$ p_m $的整数，是订单$ 2M $ $ $ 2M $的交织操作员。更具体地说，我们表明，当$ n \ geq 2m+4 $时，$ c^1（\ mathbb {s}^n）$中的每个正函数都可以在$ c^1中近似于$ c^1（\ mathbb {s}^n）$ norm y n norm a c^1 in c^1（\ mathbb {\ mathbb {\ mathbb {\ mathbb {奇异集的解决方案是一个点。此外，可以将这种解决方案构建为在其奇点附近任意大的。 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.
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.
