Non-degeneracy of the bubble in a fractional and singular 1D Liouville equation

TL;DR

This paper proves the non-degeneracy of solutions to a fractional singular Liouville equation on the real line, using conformal transformations and Steklov eigenvalue analysis to establish eigenvalue simplicity.

Contribution

It introduces a novel approach by transforming the linearized problem into a Steklov eigenvalue problem to analyze solution non-degeneracy.

Findings

Eigenvalue simplicity established for the linearized problem

Non-degeneracy of solutions proven in the fractional singular setting

Method applicable to equations on the entire real line with singular terms

Abstract

We prove the non-degeneracy of solutions to a fractional and singular Liouville equation defined on the whole real line in presence of a singular term. We use conformal transformations to rewrite the linearized equation as a Steklov eigenvalue problem posed in a bounded domain, which is defined either by an intersection or a union of two disks. We conclude by proving the simplicity of the corresponding eigenvalue.