Classification of stable solutions to a non-local Gelfand-Liouville equation

TL;DR

This paper investigates the stability of solutions to a non-local Gelfand-Liouville equation, establishing non-existence of finite Morse index solutions under certain instability conditions for all fractional orders s in (0,1).

Contribution

It provides a classification of stable solutions to a non-local Gelfand-Liouville problem, extending understanding of solution stability for fractional Laplacian equations.

Findings

Non-existence of finite Morse index solutions under instability of a specific singular solution.

Characterization of stability conditions for solutions across all s in (0,1).

Extension of classical results to non-local fractional Laplacian equations.

Abstract

We study finite Morse index solutions to the non-local Gelfand-Liouville problem $$ (-\Delta)^su=e^u\quad\mathrm{in}\quad \mathbb{R}^n,$$ for every $s\in(0,1)$ and $n>2s$. Precisely, we prove non-existence of finite Morse index solutions whenever the singular solution $$u_{n,s}(x)=-2s\log|x|+\log \left(2^{2s}\frac{\Gamma(\frac{n}{2})\Gamma(1+s)}{\Gamma(\frac{n-2s}{2})}\right)$$ is unstable.