On finite Morse index solutions of higher order fractional elliptic equations

Abstract

We consider sign-changing solutions of the equationwhere n ≥ 1, λ > 0, p > 1 and 1 < s ≤ 2. The main goal of this work is to analyze the influence of the linear term λu, 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 R n . We first derive a monotonicity formula for our equation. After that, we provide integral estimate from stability which combined with Pohozaevtype identity to obtain nonexistence results in the subcritical case with the restrictive condition. 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 R n . Through this approach we give a complete classification of stable solutions for all p > 1. Moreover, for the case 0 < s ≤ 1, finite Morse index solutions are classified in [19,25].