  Nonexistence results for positive supersolutions of the equation
$$-Lu=u^p\quad\text{in $\mathbb R^N_+$}$$ are obtained, $-L$ being any
symmetric and stable linear operator, positively homogeneous of degree $2s$,
$s\in(0,1)$, whose spectral measure is absolutely continuous and positive only
in a relative open set of the unit sphere of $\mathbb R^N$. The results are
sharp: $u\equiv 0$ is the only nonnegative supersolution in the subcritical
regime $1\leq p\leq\frac{N+s}{N-s}\,$, while nontrivial supersolutions exist,
at least for some specific $-L$, as soon as $p>\frac{N+s}{N-s}$. \\ The
arguments used rely on a rescaled test function's method, suitably adapted to
such nonlocal setting with weak diffusion; they are quite general and also
employed to obtain Liouville type results in the whole space.
