The Pohozaev Identity for the Fractional Laplacian

In this paper we prove the Pohozaev identity for the semilinear Dirichlet problem $${(-\Delta)^s u =f(u)}$$(-Δ)su=f(u) in $${\Omega, u\equiv0}$$Ω,u≡0 in $${{\mathbb R}^n\backslash\Omega}$$Rn\Ω . Here, $${s\in(0,1)}$$s∈(0,1) , (−Δ)s is the fractional Laplacian in $${\mathbb{R}^n}$$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 $${u/\delta^s|_{\Omega}}$$u/δs|Ω is Cα up to the boundary ∂Ω, where δ(x) = dist(x,∂Ω). In the fractional Pohozaev identity, the function $${u/\delta^s|_{\partial\Omega}}$$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.