The Best Constant, the Nonexistence of Extremal Functions and Related Results for an Improved Hardy-Sobolev Inequality

TL;DR

This paper determines the optimal constant and extremal functions for an improved Hardy-Sobolev inequality, revealing its equivalence to Sobolev inequality and connecting it to related functional inequalities.

Contribution

It establishes the best constant and existence of extremal functions for an improved Hardy-Sobolev inequality, and links it to Sobolev and Caffarelli-Kohn-Nirenberg inequalities.

Findings

Identified the best constant for the inequality

Proved the existence of extremal functions

Connected the inequality to Sobolev and related inequalities

Abstract

We present the best constant and the existence of extremal functions for an Improved Hardy-Sobolev inequality. We prove that, under a proper transformation, this inequality is equivalent to the Sobolev inequality in $\mathbb{R}^N$. We also discuss the connection of the related functional spaces and as a result we obtain some Caffarelli - Kohn - Nirenberg inequalities. Our starting point is the existence of a minimizer for the Bliss' inequality and the indirect dependence of the Hardy inequality at the origin.