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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 infinitesimalgenerator of the censored symmetric $2\alpha$-stable process in $\Omega$.Probability theory asserts that the censored $2\alpha$-stable process can notapproach the boundary when $\alpha\in(0,\frac12]$.  For $\alpha\in (0,\frac12]$, our purpose in this article is to show thatnon-existence of solutions bounded from above or bounded from below for theparticular Poisson problem $$  (-\Delta)^\alpha_\Omega u= 1 \quad {\rm in}\ \, \, \Omega  $$ and non-existence of nonnegative nontrivial solutions of the Lane-Emdenequation $$  (-\Delta)^\alpha_\Omega u=u^p\quad {\rm in}\ \, \Omega,\qquad u=0\quad {\rmon}\ \partial\Omega.  $$

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