Classification of Stable Solutions to a Non-Local Gelfand–Liouville Equation. (8th September 2020)

• Record Type: Journal Article
• Authors: Hyder, Ali; Yang, Wen
• Is Part Of: International mathematics research notices. Volume 2022:Number 7(2022)
• Page Start: 5219
• Page End: 5255
• Publication Date: 2020-09-08
• Subjects: Mathematics -- Periodicals; 510
• DOI: 10.1093/imrn/rnaa236
• Languages: English
• ISSNs: 1073-7928

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.