Liouville results for semilinear integral equations with conical diffusion

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