On stable and finite Morse index solutions of the nonlocal Hénon-Gelfand-Liouville equation

Abstract

We consider the nonlocal Hénon-Gelfand-Liouville problem $$ (-Δ)^s u = |x|^a e^u\quad\mathrm{in}\quad \mathbb R^n, $$ for every $s\in(0,1)$, $a>0$ and $n>2s$. We prove a monotonicity formula for solutions of the above equation using rescaling arguments. We apply this formula together with blow-down analysis arguments and technical integral estimates to establish non-existence of finite Morse index solutions when $$\dfrac{Γ(\frac n2)Γ(s)}{Γ(\frac{n-2s}{2})}\left(s+\frac a2\right)> \dfrac{Γ^2(\frac{n+2s}{4})}{Γ^2(\frac{n-2s}{4})}.$$