Nonexistence results for a class of fractional elliptic boundary value problems

In this paper we study a class of fractional elliptic problems of the form { (-Delta)(s)u = f(x, u) in Omega, u = 0 in R-N\Omega, where s is an element of (0, 1). We prove nonexistence of positive solutions when Omega is star-shaped and is supercritical. We also derive a nonexistence result for subcritical f in some unbounded domains. The argument relics 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. (C) 2012 Elsevier Inc. All rights reserved.