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

Abstract

We classify finite Morse index solutions of the following nonlocal Lane-Emden equation $$(-\Delta)^{s} u=|u|^{p-1} u\quad {\Bbb R}^n$$ for $1<s<2$ via a novel monotonicity formula. For local cases $s=1$ and $s=2$ this classification was provided by Farina in 2007 and D{\'a}vila, Dupaigne, Wang, and Wei in 2014, respectively. Moreover, for the nonlocal case $0<s<1$ finite Morse index solutions are classified by D{\'a}vila, Dupaigne, and Wei in their 2014 preprint.