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 equivalent to 0 in R-n\Omega. Here, s is an element of (0, 1), (-Delta)(s) is the fractional Laplacian in R-n, and Omega is a bounded C-1,C-1 domain. To establish the identity we use, among other things, that if u is a bounded solution then u/delta(s)vertical bar(Omega) is C-alpha up to the boundary partial derivative Omega, where delta(x) = dist(x, partial derivative Omega). In the fractional Pohozaev identity, the function u/delta(s)vertical bar(partial derivative Omega) plays the role that partial derivative u/partial derivative nu plays in the classical one. Surprisingly, from a nonlocal problem we obtain an identity with a boundary term (an integral over partial derivative Omega) which is completely local. As an application of our identity, we deduce the nonexistence of nontrivial solutions in star-shaped domains for supercritical nonlinearities.