论文标题
Liouville定理用于椭圆方程，涉及区域分数Laplacian，订单为$（0，\，1/2] $
Liouville theorems for elliptic equations involving regional fractional Laplacian with order in $(0,\,1/2]$
论文作者
论文摘要
在本文中，考虑了$ \ mathbb {r}^n $（$ n \ geq 2 $）中有限的开放域中的一些椭圆方程，并考虑了$ c^2 $ boundard $ \partialΩ$。该问题是由区域分数拉普拉斯（Laplacian）驱动的，该区域是$ω$中审查的对称$2α$稳定过程的无限发电机。概率理论断言，当$α\ in（0，\ frac12] $）$α\时，审查的$2α$稳定过程无法接近边界。 对于$α\ in（0，\ frac12] $，我们在本文中的目的是表明解决方案的不存在从上方界定或从下方界限的特定泊松问题$$ （-Δ）^α_ΩU= 1 \ Quad {\ rm in} \ \，\，ω $$和不存在的非负式解决方案$$ $$ （-Δ）^α_Ω $$
In this paper, some elliptic equation in a bounded open domain in $\mathbb{R}^N$ ($N\geq 2$) with $C^2$ boundary $\partialΩ$ is considered. The problem is driven by the regional fractional Laplacian, the infinitesimal generator of the censored symmetric $2α$-stable process in $Ω$. Probability theory asserts that the censored $2α$-stable process can not approach the boundary when $α\in(0,\frac12]$. For $α\in (0,\frac12]$, our purpose in this article is to show that non-existence of solutions bounded from above or bounded from below for the particular Poisson problem $$ (-Δ)^α_Ωu= 1 \quad {\rm in}\ \, \, Ω $$ and non-existence of nonnegative nontrivial solutions of the Lane-Emden equation $$ (-Δ)^α_Ωu=u^p\quad {\rm in}\ \, Ω,\qquad u=0\quad {\rm on}\ \partialΩ. $$
