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On the eigenvalues of the $p$&$q$-fractional Laplacian

Sabri Bahrouni, Hichem Hajaiej and Linjie Song
Vol. 6 (2024), No. 4, 735–759
DOI: 10.2140/tunis.2024.6.735
Abstract

We consider the eigenvalue problem for the fractional p&q-Laplacian

{ (−Δ)psu + μ(−Δ)qsu + |u|p−2u + μ|u|q−2u = λV (x)|u|p−2u in Ω, u = 0 in ℝN∖Ω,

where Ω is an open, bounded, and possibly disconnected domain, λ ∈ ℝ, 1 < q < p < N∕s, μ > 0 with a weight function in L∞(Ω) that is allowed no change of sign. We show that the problem has a continuous spectrum. Moreover, our result reveals a discontinuity property for the spectrum as the parameter μ goes to 0+. In addition, a stability property of eigenvalues as s → 1− is established.

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Keywords
fractional $p$&$q$-Laplacian, eigenvalues, continuous spectrum, stability of eigenvalues
Mathematical Subject Classification
Primary: 35P30, 47J10
Milestones
Received: 17 December 2023
Accepted: 30 April 2024
Published: 18 December 2024
Authors
Sabri Bahrouni
University of Monastir
Mathematics Department
Faculty of Sciences
University of Monastir
Monastir
Tunisia
Hichem Hajaiej
Department of Mathematics
California State University
Los Angeles, CA
United States
Linjie Song
Institute of Mathematics
AMSS, Academia Sinica
Beijing
China
© 2024 MSP (Mathematical Sciences Publishers).