Nonexistence results for positive supersolutions of the equation \(-Lu=u^p\quad \hbox { in}\ \mathbb {R}^N_+\) 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.
