Summary: In this paper we study a class of fractional elliptic problems of the form
where . We prove nonexistence of positive solutions when is star-shaped and is supercritical. We also derive a nonexistence result for subcritical in some unbounded domains. The argument relies on the method of moving spheres applied to a reformulated problem using the L. Caffarelli and L. Silvestre extension [Commun. Partial Differ. Equations 32, No. 8, 1245–1260 (2007; Zbl 1143.26002)] of a solution of the above problem.