Mass and asymptotics associated to fractional Hardy-Schrödinger operators in critical regimes

• Published in: Communications in partial differential equations Vol. 43; no. 6; pp. 859 - 892
• Main Authors: Ghoussoub, Nassif, Robert, Frédéric, Shakerian, Shaya, Zhao, Mingfeng
• Format: Journal Article
• Language: English
• Published: Philadelphia Taylor & Francis 03-06-2018; Taylor & Francis Ltd
• Subjects: Analysis of PDEs; Boundary value problems; Critical exponent; Domains; Eigenvalues; fractional Laplacian; Hardy inequalities; Linear equations; Mathematics; nonlocal operators; Operators (mathematics)
• ISSN: 0360-5302; 1532-4133
• DOI: 10.1080/03605302.2018.1476528

We consider linear and non-linear boundary value problems associated to the fractional Hardy-Schrödinger operator
on domains of
containing the singularity 0, where
and
, the latter being the best constant in the fractional Hardy inequality on
. We tackle the existence of least-energy solutions for the borderline boundary value problem
on Ω, where
and
is the critical fractional Sobolev exponent. We show that if γ is below a certain threshold γ
crit
, then such solutions exist for all
, the latter being the first eigenvalue of
. On the other hand, for
, we prove existence of such solutions only for those λ in
for which the domain Ω has a positive fractional Hardy-Schrödinger mass
. This latter notion is introduced by way of an invariant of the linear equation
on Ω.