  In this paper we prove the Pohozaev identity for the semilinear Dirichlet
problem $(-\Delta)^s u = f(u)$ in $\Omega$, $u \equiv 0$ in $\mathbb
R^n\setminus\Omega$. Here, $s\in(0,1)$, $(-\Delta)^s$ is the fractional
Laplacian in $\mathbb R^n$, and $\Omega$ 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 $C^\alpha$ up to the boundary
$\partial\Omega$, where $\delta(x)={\rm dist}(x,\partial\Omega)$. In the
fractional Pohozaev identity, the function $u/\delta^s|_{\partial\Omega}$ plays
the role that $\partial u/\partial\nu$ plays in the classical one.
Surprisingly, from a nonlocal problem we obtain an identity with a boundary
term (an integral over $\partial\Omega$) which is completely local.
  As an application of our identity, we deduce the nonexistence of nontrivial
solutions in star-shaped domains for supercritical nonlinearities.
