The Pohozaev identity for the fractional Laplacian

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 $\delta (x)=\mathrm{dist}(x,\partial \Omega )$. 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.

The sign of the boundary term in Theorem 1.9 has been corrected

References in corpus (1)

• Nonexistence results for a class of fractional elliptic boundary value problems

Cited by in corpus (1)

• Existence and symmetry results for a Schrödinger type problem involving the fractional Laplacian