  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 $$
  (-\Delta)^\alpha_\Omega u=u^p\quad {\rm in}\ \, \Omega,\qquad u=0\quad {\rm
on}\ \partial\Omega.
  $$
