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 $\mathbb{R}^N$ ($N\geq 2$) 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 $\alpha\in(0,\frac12]$. For $\alpha\in (0,\frac12]$, 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 $$ (-\Delta)^\alpha_\Omega u= 1 \quad {\rm in}\ \, \, \Omega $$ and non-existence of nonnegative nontrivial solutions of the Lane-Emden equation $$ (-\Delta)^\alpha_\Omega u=u^p\quad {\rm in}\ \, \Omega,\qquad u=0\quad {\rm on}\ \partial\Omega. $$