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

Abstract

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 γcrit, 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.