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 $u\mathrm{/}{d}^s{\mathrm{\mid }}_{\mathrm{\partial }\mathrm{\Omega }}$ plays the role that $\mathrm{\partial }u\mathrm{/}\mathrm{\partial }\nu$ plays in the second order case. Here, $u$ is any solution to $Lu=f(x,u)$ in $\mathrm{\Omega }$, with $u=0$ in ${\mathbb{R}}^n\setminus \mathrm{\Omega }$, and $d$ is the distance to $\mathrm{\partial }\mathrm{\Omega }$.