Classification of the stable solution to the fractional $2<s<3$ Lane-Emden equation

TL;DR

This paper classifies all stable solutions, including positive and sign-changing ones, to a nonlocal fractional Lane-Emden equation for the range 2<s<3, expanding understanding of solution behaviors in nonlocal PDEs.

Contribution

It provides a comprehensive classification of stable solutions to the fractional Lane-Emden equation for 2<s<3, including non-radial and sign-changing solutions, which was previously unexplored.

Findings

Complete classification of stable solutions for 2<s<3

Includes positive and sign-changing solutions

Addresses radial and non-radial cases

Abstract

We classify the stable solutions (positive or sign-changing, radial or not) to the following nonlocal Lane-Emden equation: $(-\Delta)^s u=|u|^{p-1}u$ in $\mathbb{R}^n$ for $2<s<3$.