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

We consider linear and non-linear boundary value problems associated to the fractional Hardy-Schrodinger 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 , where and is the critical fractional Sobolev exponent. We show that if is below a certain threshold (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 in for which the domain has a positive fractional Hardy-Schrodinger mass. This latter notion is introduced by way of an invariant of the linear equation on Omega.