We consider linear stable operators L of order 2s whose spectral measure is positive only in a relative open subset of the unit sphere, the aim being to present Liouville type results, in a half space, for the inequality $-Lu ≥ u^p$. In particular we will show that u≡0 is the only nonnegative solution for $1 ≤ p ≤ (N+s)/(N-s)$. The optimality of the exponent $(N+s)/(N-s)$ will also be discussed. Based on a joint work with I. Birindelli and L. Du (Sapienza Università di Roma)