Nonexistence results for a class of fractional elliptic boundary value problems

Abstract

In this paper we study a class of fractional elliptic problems of the form{(−Δ)su=f(x,u)in Ω,u=0in RN∖Ω, where s∈(0,1). We prove nonexistence of positive solutions when Ω is star-shaped and f is supercritical. We also derive a nonexistence result for subcritical f in some unbounded domains. The argument relies on the method of moving spheres applied to a reformulated problem using the Caffarelli–Silvestre extension (Caffarelli and Silvestre (2007) [11]) of a solution of the above problem.