Qualitative Properties of Singular Solutions to the Fractional Yamabe Problem

TL;DR

This paper investigates the qualitative behavior of solutions with isolated singularities to the fractional Yamabe problem, showing that such solutions have infinite Morse index using a transformation to a 1D nonlocal problem.

Contribution

It establishes that solutions with isolated singularities to the fractional Yamabe problem have infinite Morse index, advancing understanding of their stability properties.

Findings

Solutions with isolated singularities have infinite Morse index.

The proof employs an Emden Fowler type transformation.

The approach reduces the problem to a nonlocal 1D analysis.

Abstract

In this paper we are interested in the qualitative properties of the solutions to the fractional Yamabe problem in $\mathbb{R}^n$ which present an isolated singularity. In particular, we prove that the Morse index of any such solution is infinity. The proof uses a Emden Fowler type transformation, so that we can pass to a nonlocal 1D problem posed in $\mathbb{R}$.