The Pohozaev identity for the Spectral Fractional Laplacian
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arXiv
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| Format: | Preprint |
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2026
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| author | Barrios-Cubas, Itahisa Bonforte, Matteo González, María del Mar Torres-Latorre, Clara |
| author_facet | Barrios-Cubas, Itahisa Bonforte, Matteo González, María del Mar Torres-Latorre, Clara |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_16185 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | The Pohozaev identity for the Spectral Fractional Laplacian Barrios-Cubas, Itahisa Bonforte, Matteo González, María del Mar Torres-Latorre, Clara Analysis of PDEs 35J61, 35R11 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. |
| title | The Pohozaev identity for the Spectral Fractional Laplacian |
| topic | Analysis of PDEs 35J61, 35R11 |
| url | https://arxiv.org/abs/2601.16185 |