Liouville results for semilinear integral equations with conical diffusion

Nonexistence results for positive supersolutions of the equation (Formula presented.) 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≤N+sN-s, while nontrivial supersolutions exist, at least for some specific -L, as soon as p>N+sN-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.