Borderline variational problems involving fractional Laplacians and critical singularities

TL;DR

This paper investigates the attainability of the best constant in a critical fractional Hardy-Sobolev inequality involving fractional Laplacians and singularities, establishing existence of solutions for a related doubly critical PDE.

Contribution

It introduces new conditions for attainability of the best constant and proves the existence of nontrivial solutions for a fractional PDE with critical singularities.

Findings

Established conditions for the attainability of the best constant.

Proved existence of positive weak solutions for the doubly critical problem.

Extended fractional Hardy-Sobolev inequality theory to include singularities.

Abstract

We consider the problem of attainability of the best constant in the following critical fractional Hardy-Sobolev inequality: \begin{equation*} \mu_{\gamma,s}(\R^n):= \inf\limits_{u \in H^{\frac{\alpha}{2}} (\R^n)\setminus \{0\}} \frac{ \int_{\R^n} |({-}{ \Delta})^{\frac{\alpha}{4}}u|^2 dx - \gamma \int_{\R^n} \frac{|u|^2}{|x|^{\alpha}}dx }{(\int_{\R^n} \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 $\R^n$, \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 } {\R^n}\\ \hfill u&>0 & \text{in } \R^n, \end{array}\right. \end{equation*} where $2_{\alpha}^*:=\frac{2 n}{n-{\alpha}}$ is the critical $\alpha$-fractional Sobolev exponent, and $\gamma < \gamma_H:=2^\alpha \frac{\Gamma^2(\frac{n+\alpha}{4})}{\Gamma^2(\frac{n-\alpha}{4})}$, the latter being the best fractional Hardy constant on $\R^n$.