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 $L_{\gamma ,\alpha }:=(-\Delta {)}^{\frac{\alpha }{2}}-\frac{\gamma }{|x{|}^{\alpha }}$ 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 $(L_{\gamma ,\alpha }-\lambda I)u=\frac{u^{2_{\alpha }^{\star }(s)-1}}{|x{|}^s}$ on $\Omega$, where $0\le s<\alpha <n$ and $2_{\alpha }^{\star }(s)=\frac{2(n-s)}{n-\alpha }$ 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-Schrödinger 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 $\Omega$.