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

Abstract

We study finite Morse index solutions to the non-local Gelfand-Liouville problem $$(-\Delta {)}^{s}u=e^u\mathrm{in}{\mathbb{R}}^n,$$ 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.

25 pages

References in corpus (1)

• The Gel'fand problem for the biharmonic operator

Cited by in corpus (1)

• Liouville type theorems for fractional elliptic problems