Very large solutions for the fractional Laplacian: Towards a fractional Keller-Osserman condition

Abstract

We look for solutions of (-) s u + f (u) = 0 s u+f(u)=0 in a bounded smooth domain Ω, s ϵ (0,1) s\in(0,1), with a strong singularity at the boundary. In particular, we are interested in solutions which are L 1 (Ω) L 1(Ω) and higher order with respect to dist (x, Ω) s - 1 dist (x, Ω) s-1. We provide sufficient conditions for the existence of such a solution. Roughly speaking, these functions are the real fractional counterpart of large solutions in the classical setting.