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

The purpose of this paper is to study Liouville theorems for elliptic equations involving regional fractional Laplacian. The regional fractional Laplacian is the infinitesimal generator of the censored symmetric $2\alpha$-stable process in $\Omega$. Probability theory asserts that the censored $2\alpha$-stable process cannot approach the boundary when $\alpha\in(0,\frac12]$. We show the nonexistence of solutions bounded from above or bounded from below for Poisson equation $$ (-\Delta)^\alpha_\Omega u= 1 \quad {\rm in}\ \, \, \Omega $$ and nonexistence of solutions for the following Lane-Emden equation $$ \displaystyle (-\Delta)^\alpha_\Omega u=u^p\quad {\rm in}\ \, \Omega,\qquad u=0\quad {\rm on}\ \partial\Omega. $$