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

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