On stable and finite Morse index solutions of the nonlocal H\'{e}non-Gelfand-Liouville equation

TL;DR

This paper establishes non-existence results for finite Morse index solutions of a nonlocal Hénon-Gelfand-Liouville equation in \\mathbb{R}^n using a monotonicity formula, blow-down analysis, and integral estimates, under specific parameter conditions.

Contribution

It introduces a new monotonicity formula and applies blow-down analysis to prove non-existence of finite Morse index solutions for the nonlocal equation.

Findings

Non-existence of finite Morse index solutions under certain parameter conditions.

Development of a monotonicity formula for nonlocal equations.

Application of blow-down analysis and integral estimates to establish results.

Abstract

We consider the nonlocal H\'{e}non-Gelfand-Liouville problem $$ (-\Delta)^s u = |x|^a e^u\quad\mathrm{in}\quad \mathbb R^n, $$ 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 $$\dfrac{\Gamma(\frac n2)\Gamma(s)}{\Gamma(\frac{n-2s}{2})}\left(s+\frac a2\right)> \dfrac{\Gamma^2(\frac{n+2s}{4})}{\Gamma^2(\frac{n-2s}{4})}.$$