论文标题
欧几里得空间中恒定负$ q $ curvature的奇异指标
Singular metrics of constant negative $Q$-curvature in Euclidean spaces
论文作者
论文摘要
我们研究欧几里得空间中常数$ q $ curvature的单数指标$ \ mathbb {r}^n $每$ n \ geq 1 $。确切地说，我们考虑解决问题的解决方案\ [ （-Δ）^{n/2} u = -e^{nu} \ quad \ text {on} \ quad \ quad \ mathbb {r}^{n} \ backslash \ {0 \}，\ {0 \}，\]，在有限体积条件下我们根据上述方程的所有单数解根据无穷大的行为和零分类。因此，当$ n = 1,2 $时，我们表明实际上没有单数解决方案。然后改编一种变分技术，我们可以为任何$ n \ geq 3 $和$λ> 0 $获得该方程，该方程将接受具有规定的渐近行为的解决方案。这些解决方案对应于恒定负$ q $ curvature的指标，它们要么平滑，要么具有对数或多项式类型的奇异性。本文补充了关于正面$ Q $ curvature的案例的先前作品，并且还可以在非词性负$ q $ curvature案件中提高先前的结果。
We study singular metrics of constant negative $Q$-curvature in the Euclidean space $\mathbb{R}^n$ for every $n \geq 1$. Precisely, we consider solutions to the problem \[ (-Δ)^{n/2}u=-e^{nu}\quad \text{on}\quad\mathbb{R}^{n}\backslash \{0\}, \] under a finite volume condition $Λ:=\int_{\mathbb{R}^n}e^{nu}dx$. We classify all singular solutions of the above equation based on their behavior at infinity and zero. As a consequence of this, when $n=1,2$, we show that there is actually no singular solution. Then adapting a variational technique, we obtain that for any $n\geq 3$ and $Λ>0$, the equation admits solutions with prescribed asymptotic behavior. These solutions correspond to metrics of constant negative $Q$-curvature, which are either smooth or have a singularity at the origin of logarithmic or polynomial type. The present paper complements previous works on the case of positive $Q$-curvature, and also sharpens previous results in the nonsingular negative $Q$-curvature case.
