On stable solutions of the fractional Henon-Lane-Emden equation

TL;DR

This paper develops monotonicity formulas for the fractional Hénon-Lane-Emden equation and uses them to classify its stable solutions, advancing understanding of fractional nonlinear PDEs.

Contribution

It introduces new monotonicity formulas for the fractional Hénon-Lane-Emden equation and applies these to classify stable solutions, providing novel insights into fractional PDE stability.

Findings

Derived monotonicity formulas for solutions

Classified stable solutions of the fractional equation

Enhanced understanding of fractional nonlinear PDE stability

Abstract

We derive monotonicity formulae for solutions of the fractional H\'{e}non-Lane-Emden equation \begin{equation*} (-\Delta)^{s} u=|x|^a |u|^{p-1} u \ \ \ \text{in } \ \ \mathbb{R}^n, \end{equation*} when $0<s<2$, $a>0$ and $p>1$. Then, we apply these formulae to classify stable solutions of the above equation.