The Pohozaev identity for the fractional Laplacian

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