On finite Morse index solutions of higher order fractional Lane-Emden equations

TL;DR

This paper classifies finite Morse index solutions of higher order fractional Lane-Emden equations for the range 1<s<2, extending previous classifications for local and nonlocal cases.

Contribution

It provides a new classification of finite Morse index solutions for the fractional Lane-Emden equation when 1<s<2, filling a gap between known local and nonlocal cases.

Findings

Classification of solutions for 1<s<2

Extension of previous classifications to higher order fractional cases

Bridges gap between local and nonlocal solution classifications

Abstract

We classify finite Morse index solutions of the fractional Lane-Emden equation $(-\Delta)^{s} u=|u|^{p-1} u \ \ \ \mathbb{R}^n $ for $1<s<2$. For the local case, $s=1$ and $s=2$ this classification was done by Farina in [10] and Davila, Dupaigne, Wang and Wei in [8], respectively. Moreover, for the nonlocal case, $0<s<1$, finite Morse index solutions are classified by Davila, Dupaigne and Wei in [7].