Classification of Stable Solutions to a Non-Local Gelfand–Liouville Equation

Abstract

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