Mass and Asymptotics associated to Fractional Hardy-Schr\"odinger Operators in Critical Regimes

We consider linear and non-linear boundary value problems associated to the fractional Hardy-Schr\"odinger operator $L_{\gamma,\alpha}: = ({-}{ \Delta})^{\frac{\alpha}{2}}- \frac{\gamma}{|x|^{\alpha}}$ on domains of $\mathbb{R}^n$ containing the singularity $0$, where $0<\alpha<2$ and $0 \le \gamma < \gamma_H(\alpha)$, the latter being the best constant in the fractional Hardy inequality on $\mathbb{R}^n$. We tackle the existence of least-energy solutions for the borderline boundary value problem $(L_{\gamma,\alpha}-\lambda I)u= {\frac{u^{2^\star_\alpha(s)-1}}{|x|^s}}$ on $\Omega$, where $0\leq s <\alpha <n$ and $2^\star_\alpha(s)={\frac{2(n-s)}{n-{\alpha}}}$ is the critical fractional Sobolev exponent. We show that if $\gamma$ is below a certain threshold $\gamma_{crit}$, then such solutions exist for all $0<\lambda <\lambda_1(L_{\gamma,\alpha})$, the latter being the first eigenvalue of $L_{\gamma,\alpha}$. On the other hand, for $\gamma_{crit}<\gamma <\gamma_H(\alpha)$, we prove existence of such solutions only for those $\lambda$ in $(0, \lambda_1(L_{\gamma,\alpha}))$ for which the domain $\Omega$ has a positive {\it fractional Hardy-Schr\"odinger mass} $m_{\gamma, \lambda}(\Omega)$. This latter notion is introduced by way of an invariant of the linear equation $(L_{\gamma,\alpha}-\lambda I)u=0$ on $\Omega$