A note on the eigenvalues of fractional Hardy-Sobolev operator with indefinite weight

2016/07/26 by Sarika Goyal, Goyal, Sarika
Mathematics · #35A15 #35B33 #35H39 #Advanced Harmonic Analysis Research #Analysis of PDEs (math.AP) #Differential Equations and Boundary Problems #FOS: Mathematics #Nonlinear Partial Differential Equations #math.AP #msc:35A15 #msc:35B33 #msc:35H39

paper · pdf · doi:10.48550/arxiv.1607.07580

30 pages

arxiv created 2016/07/26 · openalex publication_date 2016/07/26 · arxiv updated 2016/07/27 · openalex created_date 2019/07/30 · openalex updated_date 2026/07/28

Abstract

In this article, we study the eigenvalue of nonlinear p-fractional Hardy operator (-Δ)_p^αu - μ\frac|u|^p-2u|x|^pα = λV(x) |u|^p-2u in Ω, u = 0 in ℝⁿ ∖Ω, where n>pα, p≥2, α∈(0,1), 0≤ μ<C_n,α,p and Ω is a domain in ℝⁿ with Lipschitz boundary containing 0. In particular, Ω=ℝⁿ is admitted. The weight function V may change sign and may have singular points. We also show that the least positive eigenvalue is simple and it is unique associated to a non-negative eigenfunction. Moreover, we proved that there exists a sequence of eigenvalues λ_k → ∞ as k→∞.

Citations

• Fractional p-eigenvalues
• Hitchhikerʼs guide to the fractional Sobolev spaces
• The Brezis-Nirenberg result for the fractional Laplacian
• An optimization problem for the first eigenvalue of the p-fractional laplacian
• Stability of variational eigenvalues for the fractional p-Laplacian

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• Characterizations of compactness and weighted eigenvalue problem associated with fractional Hardy-type inequalities