Borderline variational problems involving fractional Laplacians and critical singularities (1503.08193v2)
Abstract: We consider the problem of attainability of the best constant in the following critical fractional Hardy-Sobolev inequality: \begin{equation*} \mu_{\gamma,s}(\Rn):= \inf\limits_{u \in H{\frac{\alpha}{2}} (\Rn)\setminus {0}} \frac{ \int_{\Rn} |({-}{ \Delta}){\frac{\alpha}{4}}u|2 dx - \gamma \int_{\Rn} \frac{|u|2}{|x|{\alpha}}dx }{(\int_{\Rn} \frac{|u|{2_{\alpha}(s)}}{|x|{s}}dx)\frac{2}{2_{\alpha}^(s)}}, \end{equation*} where $0\leq s<\alpha<2$, $n>\alpha$, ${2_{\alpha}*(s)}:=\frac{2(n-s)}{n-{\alpha}},$ and $\gamma \in \mathbb{R}$. This allows us to establish the existence of nontrivial weak solutions for the following doubly critical problem on $\Rn$, \begin{equation*} \left{\begin{array}{lll} ({-}{ \Delta}){\frac{\alpha}{2}}u- \gamma \frac{u}{|x|{\alpha}}&= |u|{2_{\alpha}*-2} u + {\frac{|u|{2_{\alpha}*(s)-2}u}{|x|s}} & \text{in } {\Rn}\ \hfill u&>0 & \text{in } \Rn, \end{array}\right. \end{equation*} where $2_{\alpha}*:=\frac{2 n}{n-{\alpha}}$ is the critical $\alpha$-fractional Sobolev exponent, and $\gamma < \gamma_H:=2\alpha \frac{\Gamma2(\frac{n+\alpha}{4})}{\Gamma2(\frac{n-\alpha}{4})}$, the latter being the best fractional Hardy constant on $\Rn$.