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 ) s u = f ( x , u ) in Ω , u = 0 in R N ∖ 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.

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