Liouville theorems for elliptic equations involving regional fractional Laplacian with order in $(0,1/2]$

Abstract

In this paper, some elliptic equation in a bounded open domain in ${\mathbb{R}}^N$ ($N\ge 2$) with $C^2$ boundary $\partial \Omega$ is considered. The problem is driven by the regional fractional Laplacian, the infinitesimal generator of the censored symmetric $2\alpha$-stable process in $\Omega$. Probability theory asserts that the censored $2\alpha$-stable process can not approach the boundary when $\alpha \in (0,\frac{1}{2}]$. For $\alpha \in (0,\frac{1}{2}]$, 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 $$(-\Delta {)}_{\Omega }^{\alpha }u=1\mathrm{in}\Omega$$ and non-existence of nonnegative nontrivial solutions of the Lane-Emden equation $$(-\Delta {)}_{\Omega }^{\alpha }u=u^p\mathrm{in}\Omega ,u=0\mathrm{on}\partial \Omega .$$

18 pages