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The Pohozaev identity for the Spectral Fractional Laplacian

Itahisa Barrios-Cubas, Matteo Bonforte, María del Mar González, Clara Torres-Latorre

TL;DR

The paper addresses the challenge of establishing a Pohozaev-type identity for the Spectral Fractional Laplacian $(-Δ|_{Ω})^{s}$ on bounded domains and uses it to derive nonexistence results for semilinear Dirichlet problems in star-shaped domains. It introduces a spectral approach that encodes the identity as a Schur product $Q^{(s)}=P^{(s)}\circ Q^{(1)}$, with a transition matrix $P^{(s)}_{jk}=\frac{λ_j^{s}-λ_k^{s}}{λ_j-λ_k}$ (and $sλ_j^{s-1}$ on degeneracies), and proves positivity via the Schur product theorem and Bochner’s theorem. The key contributions include the first Pohozaev identity for the SFL, a rigorous proof framework for the Schur-structured positivity, and explicit nonexistence criteria that cover subcritical and supercritical nonlinearities, clarifying boundary behavior distinctions from the RFL. By delineating the boundary-regularity and invariance differences between the SFL and RFL, the work broadens the nonlocal Pohozaev toolkit and provides a robust method for analyzing nonlocal semilinear elliptic equations on bounded domains.

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.

The Pohozaev identity for the Spectral Fractional Laplacian

TL;DR

The paper addresses the challenge of establishing a Pohozaev-type identity for the Spectral Fractional Laplacian on bounded domains and uses it to derive nonexistence results for semilinear Dirichlet problems in star-shaped domains. It introduces a spectral approach that encodes the identity as a Schur product , with a transition matrix (and on degeneracies), and proves positivity via the Schur product theorem and Bochner’s theorem. The key contributions include the first Pohozaev identity for the SFL, a rigorous proof framework for the Schur-structured positivity, and explicit nonexistence criteria that cover subcritical and supercritical nonlinearities, clarifying boundary behavior distinctions from the RFL. By delineating the boundary-regularity and invariance differences between the SFL and RFL, the work broadens the nonlocal Pohozaev toolkit and provides a robust method for analyzing nonlocal semilinear elliptic equations on bounded domains.

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 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.
Paper Structure (9 sections, 10 theorems, 64 equations)

This paper contains 9 sections, 10 theorems, 64 equations.

Table of Contents

  1. Introduction
  2. Main results
  3. The Spectral Fractional Laplacian and the Restricted Fractional Laplacian
  4. Known results on the Pohozaev identity
  5. Why the Ros-Oton–Serra proof strategy fails for the SFL
  6. A bit of history
  7. Proof of the Pohozaev identity for the SFL
  8. Proofs of non-existence results
  9. Acknowledgments.

Key Result

Theorem 1.1

Let $\Omega$ be a bounded $C^{1,1}$ domain, $s\in (0,1)$ and $u\in H^{\max \left\{ 2s, 3/2 \right\} }_0(\Omega)$. Define Then $Q^{(s)}$ is a positive semidefinite quadratic form whenever $Q^{(1)}$ is. In particular, Moreover, $Q^{(s)}$ can be expressed as where $\circ$ denotes the Schur (or Hadamard) product. Equivalently, in coordinates, The transition matrix $P^{(s)}_{}$ is given by

Theorems & Definitions (14)

  • Theorem 1.1: Abstract Pohozaev identity for the SFL
  • Proposition 1.2: Pohozaev inequality for semilinear equations
  • Corollary 1.3: Non-existence of solutions I
  • Corollary 1.4: Non-existence of solutions II
  • Corollary 1.5: Non-existence of solutions III
  • Theorem 1.6
  • Theorem 1.7: Proposition 1.6 of ros2014pohozaev
  • Theorem 1.8: Theorem 1.1 of ros2014pohozaev
  • Theorem 2.1
  • Theorem 2.2
  • ...and 4 more