On finite Morse index solutions of higher order fractional elliptic equations

We consider sign-changing solutions of the equation

$ (-{\Delta})^s u+\lambda u = |u|^{p-1}u \; \mbox{in}\; \mathbb R^n, $

where $ n\geq 1 $, $ \lambda>0 $, $ p>1 $ and $ 1<s\leq2 $. The main goal of this work is to analyze the influence of the linear term $ \lambda u $, in order to classify stable solutions possibly unbounded and sign-changing. We prove Liouville type theorems for stable solutions or solutions which are stable outside a compact set of $ \mathbb R^n $. We first derive a monotonicity formula for our equation. After that, we provide integral estimate from stability which combined with Pohozaev-type identity to obtain nonexistence results in the subcritical case with the restrictive condition $ |u|_{L^{\infty}( \mathbb R^n)}^{p-1}< \frac{\lambda (p+1) }{2} $. The supercritical case needs more involved analysis, motivated by the monotonicity formula, we then reduce the nonexistence of nontrivial entire solutions which are stable outside a compact set of $ \mathbb R^n $. Through this approach we give a complete classification of stable solutions for all $ p>1 $. Moreover, for the case $ 0<s\leq1 $, finite Morse index solutions are classified in [19,25].