A generalized fractional Pohozaev identity and applications

TL;DR

This paper develops a generalized fractional Pohozaev identity and explores its applications to nonexistence results, eigenvalue derivatives, and eigenvalue simplicity for the fractional Laplacian.

Contribution

It introduces a broad fractional Pohozaev identity framework and applies it to key problems in fractional PDEs, including eigenvalue analysis and nonexistence results.

Findings

Proves a generalized fractional Pohozaev identity.

Derives a Hadamard formula for fractional Laplacian eigenvalues.

Shows simplicity of radial eigenvalues in radial domains.

Abstract

We prove a fractional Pohozaev type identity in a generalized framework and discuss its applications. Specifically, we shall consider applications to nonexistence of solutions in the case of supercritical semilinear Dirichlet problems and regarding a Hadamard formula for the derivative of Dirichlet eigenvalues of the fractional Laplacian with respect to domain deformations. We also derive the simplicity of radial eigenvalues in the case of radial bounded domains and apply the Hadamard formula to this case.