Mass and asymptotics associated to fractional Hardy–Schrödinger operators in critical regimes. Issue 6 (3rd June 2018)

• Record Type: Journal Article
• Authors: Ghoussoub, Nassif; Robert, Frédéric; Shakerian, Shaya; Zhao, Mingfeng
• Is Part Of: Communications in partial differential equations. Volume 43:Issue 6(2018)
• Page Start: 859
• Page End: 892
• Publication Date: 2018-06-03
• Subjects: Critical exponent -- fractional Laplacian -- Hardy inequalities -- nonlocal operators; Differential equations, Partial -- Periodicals; 515.353
• DOI: 10.1080/03605302.2018.1476528
• Languages: English
• ISSNs: 0360-5302

Abstract

Abstract: We consider linear and non-linear boundary value problems associated to the fractional Hardy-Schrödinger operatorL γ, α : = ( - Δ ) α 2 - γ | x | α on domains ofℝ n containing the singularity 0, where0 < α < 2 and0 ≤ γ < γ H ( α ), the latter being the best constant in the fractional Hardy inequality onℝ n . We tackle the existence of least-energy solutions for the borderline boundary value problem( L γ, α - λ I ) u = u 2 α * ( s ) - 1 | x | s on Ω, where0 ≤ s < α < n and2 α * ( s ) = 2 ( n - s ) n - α is the critical fractional Sobolev exponent. We show that if γ is below a certain threshold γ crit, then such solutions exist for all0 < λ < λ 1 ( L γ, α ), the latter being the first eigenvalue ofL γ, α . On the other hand, forγ crit < γ < γ H ( α ), we prove existence of such solutions only for those λ in( 0, λ 1 ( L γ, α ) ) for which the domain Ω has a positive fractional Hardy–Schrödinger mass m γ, λ ( Ω ) . This latter notion is introduced by way of an invariant of the linear equation( L γ, α - λ I ) u = 0 on Ω.