The Pohozaev identity for the fractional laplacian

Abstract. In this paper we prove the Pohozaev identity for the semilinear Dirich-let problem (−∆)su = f(u) in Ω, u ≡ 0 in Rn\Ω. Here, s ∈ (0, 1), (−∆)s is the fractional Laplacian in Rn, and Ω is a bounded C1,1 domain. To establish the identity we use, among other things, that if u is a bounded solution then u/δs|Ω is Cα up to the boundary ∂Ω, where δ(x) = dist(x, ∂Ω). In the fractional Pohozaev identity, the function u/δs|∂Ω plays the role that ∂u/∂ν plays in the classical one. Surprisingly, from a nonlocal problem we obtain an identity with a boundary term (an integral over ∂Ω) which is completely local. As an application of our identity, we deduce the nonexistence of nontrivial solutions in star-shaped domains for supercritical nonlinearities