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Nonexistence results for a class of fractional elliptic boundary value problems. (English) Zbl 1260.35050

Summary: In this paper we study a class of fractional elliptic problems of the form

(-Δ) s u=f(x,u)inΩ,u=0in 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 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.

35J60Nonlinear elliptic equations
35A01Existence problems for PDE: global existence, local existence, non-existence
35J70Degenerate elliptic equations
26A33Fractional derivatives and integrals (real functions)
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