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arXiv · 2601.16185

The Pohozaev identity for the Spectral Fractional Laplacian

Itahisa Barrios-Cubas·Matteo Bonforte·María del Mar González·Clara Torres-Latorre
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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 $(-Δ|_Ω)^su = f(u)$ 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.

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Itahisa Barrios-Cubas, Matteo Bonforte, María del Mar González, Clara Torres-Latorre. 2026-01-22. The Pohozaev identity for the Spectral Fractional Laplacian. https://arxiv.org/abs/2601.16185

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