Pohozaev identities for anisotropic integro-differential operators

Abstract

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