Liouville results for semilinear integral equations with conical diffusion

Nonexistence results for positive supersolutions of the equation $$-Lu=u^p\text{in }{\mathbb{R}}_{+}^N$$ 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 $1\le p\le \frac{N+s}{N-s}$, 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.