The Pohozaev Identity for the Fractional Laplacian

Abstract

In this paper we prove the Pohozaev identity for the semilinear Dirichlet problem ${(-\Delta)^s u =f(u)}$ in ${\Omega, u\equiv0}$ in ${{\mathbb R}^n\backslash\Omega}$ . Here, ${s\in(0,1)}$ , (−Δ)^s is the fractional Laplacian in ${\mathbb{R}^n}$ , and Ω is a bounded C ^1,1 domain. To establish the identity we use, among other things, that if u is a bounded solution then ${u/\delta^s|_{\Omega}}$ is C ^α up to the boundary ∂Ω, where δ(x) = dist(x,∂Ω). In the fractional Pohozaev identity, the function ${u/\delta^s|_{\partial\Omega}}$ 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.