Pohozaev identities for anisotropic integro-differential operators

We establish Pohozaev identities and integration by parts type formulas for anisotropic integro-differential operators of order $2s$, with $s\in(0,1)$. These identities involve local boundary terms, in which the quantity $u/d^s|_{\partialΩ}$ plays the role that $\partial u/\partialν$ plays in the second order case. Here, $u$ is any solution to $Lu=f(x,u)$ in $Ω$, with $u=0$ in $\mathbb R^n\setminusΩ$, and $d$ is the distance to $\partialΩ$.

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