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

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

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