The Pohozaev identity for the Spectral Fractional Laplacian

TL;DR

This paper establishes a Pohozaev identity for the Spectral Fractional Laplacian, enabling non-existence results for certain nonlinear problems in star-shaped domains, using a novel spectral approach.

Contribution

It introduces the first Pohozaev identity for the Spectral Fractional Laplacian, differing from prior non-local operators, through a new spectral method exploiting quadratic structures.

Findings

Derived a Pohozaev identity for SFL

Proved non-existence results for semilinear problems

Developed a spectral approach based on eigenvalues

Abstract

In this paper, we prove a Pohozaev identity for the Spectral Fractional Laplacian (SFL). This identity allows us to establish non-existence results for the semilinear Dirichlet problem $(-\Delta|_{\Omega})^su = f(u)$ in star-shaped domains. The first such identity for non-local operators was established by Ros-Oton and Serra in 2014 for the Restricted Fractional Laplacian (RFL). However, the SFL differs fundamentally from the RFL, and the integration by parts strategy of Ros-Oton and Serra cannot be applied. Instead, we develop a novel spectral approach that exploits the underlying quadratic structure. Our main result expresses the identity as a Schur product of the classical Pohozaev quadratic form and a transition matrix that depends on the eigenvalues of the Laplacian and the fractional exponent.