The Pohozaev identity for the fractional Laplacian

TL;DR

This paper establishes a Pohozaev identity for the fractional Laplacian in bounded domains, revealing a boundary integral relation that aids in proving nonexistence of solutions for certain nonlinearities.

Contribution

The paper proves a new Pohozaev identity for the fractional Laplacian with boundary terms, extending classical results to nonlocal operators.

Findings

Derived the Pohozaev identity for fractional Laplacian in $C^{1,1}$ domains.

Showed the boundary term is local and analogous to classical boundary derivatives.

Applied the identity to prove nonexistence of solutions in star-shaped domains for supercritical nonlinearities.

Abstract

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