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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 \[ \begin{cases} (-\Delta)^s u = f(x,u) &\text{in }\Omega, \\ u = 0 & \text{in } \mathbb R^N \backslash \Omega, \end{cases} \] where \(s\in(0,1)\). We prove nonexistence of positive solutions when \(\Omega\) 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.

MSC:
35J60 Nonlinear elliptic equations
35A01 Existence problems for PDEs: global existence, local existence, non-existence
35J70 Degenerate elliptic equations
26A33 Fractional derivatives and integrals
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