Liouville results for semilinear integral equations with conical diffusion

Nonexistence results for positive supersolutions of the equation $$-Lu=u^p\quad \hbox { in}\ \mathbb {R}^N_+$$ - L u = u p in R + N are obtained, $$-L$$ - L being any symmetric and stable linear operator, positively homogeneous of degree 2s, $$s\in (0,1)$$ s ∈ ( 0 , 1 ) , whose spectral measure is absolutely continuous and positive only in a relative open set of the unit sphere of $$\mathbb {R}^N$$ R N . The results are sharp: $$u\equiv 0$$ u ≡ 0 is the only nonnegative supersolution in the subcritical regime $$1\le p\le \frac{N+s}{N-s}\,$$ 1 ≤ p ≤ N + s N - s , while nontrivial supersolutions exist, at least for some specific $$-L$$ - L , as soon as $$p>\frac{N+s}{N-s}$$ p > 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.