ON FINITE MORSE INDEX SOLUTIONS OF HIGHER ORDER FRACTIONAL ELLIPTIC EQUATIONS

We consider sign-changing solutions of the equation (-Delta)(s)u + lambda u = vertical bar u vertical bar(p-1)u in R-n, where n >= 1, lambda > 0, p > 1 and 1 < s <= 2. 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 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 vertical bar u vertical bar(p-1)(L infinity)((Rn)) < 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 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].