Qualitative properties of singular solutions to fractional elliptic equations
Qualitative properties of singular solutions to fractional elliptic equations
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DOI：
10.1017/prm.2021.52
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发表时间：
2021-09
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影响因子：
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通讯作者：
Shuibo Huang;Zhitao Zhang;Zhisu Liu
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文献类型：
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作者：
Shuibo Huang;Zhitao Zhang;Zhisu Liu
本文利用运动球法、Caffarelli-Silvestre推广公式和爆破分析，研究了分数阶椭圆型方程非负解的局部性质 \begin{align*} (-\Delta)^{\alpha} u =f(u),~~ x\in \Omega\backslash \Gamma, \end{align*} 在哪里 $0<\alpha <1$， $\Omega = \mathbb {R}^{N}$ 或 $\Omega$ 是光滑有界域， $\Gamma$ 的奇异子集是 $\Omega$ 分数容量为0时， $f(t)$ 是局部有界且正的吗 $t\in [0,\,\infty )$，和 $f(t)/t^{({N+2\alpha })/({N-2\alpha })}$ 是不变的 $t$ 对于大型 $t$，而不是每一个 $t>0$． 我们的主要结果是解满足估计 \begin{align*} f(u(x))/ u(x)\leq C d(x,\Gamma)^{{-}2\alpha}. \end{align*} 这一估计甚至对于 $\Gamma =\{0\}$． 作为应用，我们导出了球面哈纳克不等式的解，渐近对称，圆柱对称。
In this paper, by the moving spheres method, Caffarelli-Silvestre extension formula and blow-up analysis, we study the local behaviour of nonnegative solutions to fractional elliptic equations \begin{align*} (-\Delta)^{\alpha} u =f(u),~~ x\in \Omega\backslash \Gamma, \end{align*} where $0<\alpha <1$, $\Omega = \mathbb {R}^{N}$ or $\Omega$ is a smooth bounded domain, $\Gamma$ is a singular subset of $\Omega$ with fractional capacity zero, $f(t)$ is locally bounded and positive for $t\in [0,\,\infty )$, and $f(t)/t^{({N+2\alpha })/({N-2\alpha })}$ is nonincreasing in $t$ for large $t$, rather than for every $t>0$. Our main result is that the solutions satisfy the estimate \begin{align*} f(u(x))/ u(x)\leq C d(x,\Gamma)^{{-}2\alpha}. \end{align*} This estimate is new even for $\Gamma =\{0\}$. As applications, we derive the spherical Harnack inequality, asymptotic symmetry, cylindrical symmetry of the solutions.