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

Abstract

We consider the nonlocal Hénon-Gelfand-Liouville problem $$(-\Delta {)}^{s}u=|x{|}^a{e}^u\mathrm{in}{\mathbb{R}}^n,$$ 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 $$\frac{\Gamma (\frac{n}{2})\Gamma (s)}{\Gamma (\frac{n-2s}{2})}\left(s+\frac{a}{2}\right)>\frac{{\Gamma }^2(\frac{n+2s}{4})}{{\Gamma }^2(\frac{n-2s}{4})}.$$

24 pages. Comments are welcome. arXiv admin note: text overlap with arXiv:2003.03071

References in corpus (2)

• Classification of finite Morse index solutions of higher-order Gelfand-Liouville equation
• On stable and finite Morse index solutions of the fractional Toda system