Complete classification and nondegeneracy of minimizers for the fractional Hardy-Sobolev inequality, and applications

TL;DR

This paper thoroughly classifies minimizers for the fractional Hardy-Sobolev inequality, proves their nondegeneracy, and explores existence, symmetry, and qualitative properties of solutions to related equations.

Contribution

It provides a complete classification of minimizers and establishes nondegeneracy results, advancing understanding of fractional Hardy-Sobolev inequalities.

Findings

Nondegeneracy of ground state solutions

Existence of solutions to perturbed equations

Symmetry properties of solutions

Abstract

We study linear and non-linear equations related to the fractional Hardy--Sobolev inequality. We prove nondegeneracy of ground state solutions to the basic equation and investigate existence and qualitative properties, including symmetry of solutions to some perturbed equations.