Classification of Stable Solutions to a Non-Local Gelfand–Liouville Equation. (8th September 2020) - Hyder, Ali; Yang, Wen
Abstract: We study finite Morse index solutions to the non-local Gelfand–Liouville problem $$\begin{align*}& (-Δ)^su=e^u\quadin\quad{{\mathbb{R}}^n}, \end{align*}$$ for every $s\in (0, 1)$ and $n>2s$ . Precisely, we prove non-existence of finite Morse index solutions whenever the singular solution $$\begin{align*} &u_{n, s}(x)=-2s\log|x|+\log \left(2^{2s}\frac{Γ(\frac{n}{2})Γ(1+s)}{Γ(\frac{n-2s}{2})}\right)\end{align*}$$ is unstable.