The Pohozaev identity for the fractional Laplacian

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

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