In this paper, we study the asymptotic behavior of positive solutions of the fractional Hardy-Hénon equation${(-\mathrm{\Delta })}^{\sigma }u=|x{|}^{\alpha }{u}^p\mathrm{\text{in}}{B}_1\backslash \{0\}$ with an isolated singularity at the origin, where $\sigma \in (0,1)$ and the punctured unit ball $B_1\backslash \{0\}\subset {\mathbb{R}}^n$ with $n\ge 2$. When $-2\sigma <\alpha <2\sigma$ and $\frac{n+\alpha }{n-2\sigma }<p<\frac{n+2\sigma }{n-2\sigma }$, we give a classification of isolated singularities of positive solutions, and in particular, this implies sharp blow up estimates of singular solutions. Further, we describe the precise asymptotic behavior of solutions near the singularity. More generally, we classify isolated boundary singularities and describe the precise asymptotic behavior of singular solutions for a relevant degenerate elliptic equation with a nonlinear Neumann boundary condition. These results parallel those known for the Laplacian counterpart proved by Gidas and Spruck (1981) [21], but the methods are very different, since the ODEs analysis is a missing ingredient in the fractional case. Our proofs are based on a monotonicity formula, combined with blow up (down) arguments, Kelvin transformation and uniqueness of solutions of related degenerate equations on ${\mathbb{S}}_{+}^n$. We also investigate isolated singularities located at infinity of fractional Hardy-Hénon equations.

在本文中，我们研究了分数Hardy-Hénon方程正解的渐近行为${（-\mathrm{\Delta }）}^{\sigma }ü=|X{|}^{\alpha }{ü}^p\mathrm{\text{在}}{乙}_{\mathrm{1个}}\backslash \{0\}$ 在原点具有孤立的奇点 $\sigma \in （0，\mathrm{1个}）$ 和刺破的单位球 ${乙}_{\mathrm{1个}}\backslash \{0\}\subset {\mathbb{[R}}^{ñ}$ 与 $ñ\ge 2$。什么时候$-2\sigma <\alpha <2\sigma$ 和 $\frac{ñ+\alpha }{ñ-2\sigma }<p<\frac{ñ+2\sigma }{ñ-2\sigma }$，我们给出了正解的孤立奇异点的分类，尤其是，这暗示了奇异解的急剧爆炸估计。此外，我们描述了奇点附近解的精确渐近行为。更一般地，我们对孤立的边界奇点进行分类，并描述了具有非线性诺伊曼边界条件的相关退化椭圆方程的奇异解的精确渐近行为。这些结果与Gidas和Spruck（1981）[21]证明的拉普拉斯对应物的结果相似，但方法却大不相同，因为ODEs分析是分数情况下的缺失成分。我们的证明是基于单调性公式，结合了爆破（down）参数，开尔文变换和相关退化方程解的唯一性。${\mathbb{小号}}_{+}^{ñ}$。我们还研究了位于分数阶Hardy-Hénon方程无穷大处的孤立奇点。