Proceedings of the Royal Society of Edinburgh Section A: Mathematics ( IF 1.3 ) Pub Date : 2021-09-14 , DOI: 10.1017/prm.2021.52 Shuibo Huang 1 , Zhitao Zhang 2 , Zhisu Liu 3
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.
中文翻译:
分数阶椭圆方程奇异解的定性性质
本文通过动球法、Caffarelli-Silvestre 扩展公式和爆破分析,研究了分数阶椭圆方程 \begin{align*} (-\Delta)^{\alpha} u非负解的局部行为。 =f(u),~~ x\in \Omega\反斜杠 \Gamma, \end{align*} 其中$0<\alpha <1$ , $\Omega = \mathbb {R}^{N}$或$\ Omega$是一个平滑有界域,$\Gamma$是$\Omega$的一个奇异子集,小数容量为零,$f(t)$是局部有界的,并且对于$t\in [0,\,\infty ) $和$f(t)/t^{({N+2\alpha })/({N-2\alpha })} $对于大$t$,而不是每个$t>0$。我们的主要结果是解满足估计 \begin{align*} f(u(x))/ u(x)\leq C d(x,\Gamma)^{{-}2\alpha}。\end{align*} 即使对于$\Gamma =\{0\}$来说,这个估计也是新的。作为应用,我们推导出解的球面哈纳克不等式、渐近对称、圆柱对称。