Borderline variational problems involving fractional Laplacians and critical singularities

Abstract

We consider the problem of attainability of the best constant in the following critical fractional Hardy-Sobolev inequality: \begin{equation*} μ_{γ,s}(\R^n):= \inf\limits_{u \in H^{\fracα{2}} (\R^n)\setminus \{0\}} \frac{ \int_{\R^n} |({-}{ Δ})^{\fracα{4}}u|^2 dx - γ\int_{\R^n} \frac{|u|^2}{|x|^α}dx }{(\int_{\R^n} \frac{|u|^{2_α^*(s)}}{|x|^{s}}dx)^\frac{2}{2_α^*(s)}}, \end{equation*} where $0\le s<\alpha <2$, $n>\alpha$, $2_{\alpha }^{\ast }(s):=\frac{2(n-s)}{n-\alpha },$ and $\gamma \in \mathbb{R}$. This allows us to establish the existence of nontrivial weak solutions for the following doubly critical problem on ${\mathbb{R}}^n$, \begin{equation*} \left\{\begin{array}{lll} ({-}{ Δ})^{\fracα{2}}u- γ\frac{u}{|x|^α}&= |u|^{2_α^*-2} u + {\frac{|u|^{2_α^*(s)-2}u}{|x|^s}} & \text{in } {\R^n}\\ \hfill u&>0 & \text{in } \R^n, \end{array}\right. \end{equation*} where $2_{\alpha }^{\ast }:=\frac{2n}{n-\alpha }$ is the critical $\alpha$-fractional Sobolev exponent, and $\gamma <{\gamma }_H:=2^{\alpha }\frac{{\Gamma }^2(\frac{n+\alpha }{4})}{{\Gamma }^2(\frac{n-\alpha }{4})}$, the latter being the best fractional Hardy constant on ${\mathbb{R}}^n$.

24 pages, Updated version - if any - can be downloaded at http://www.birs.ca/~nassif/

References in corpus (1)

• Fractional Hardy-Sobolev elliptic problems