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

Abstract

We look for solutions of (-△)s⁢u+f⁢(u)=0{{\left(-\triangle\right)}^{s}u+f(u)=0} in a bounded smooth domain Ω, s∈(0,1){s\in(0,1)}, with a strong singularity at the boundary. In particular, we are interested in solutions which are L1⁢(Ω){L^{1}(\Omega)} and higher order with respect to dist(x,∂Ω)s-1{\operatorname{dist}(x,\partial\Omega)^{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.

Keywords

• fractional laplacian
• large solutions
• keller–osserman condition
• 35a01
• 45k05
• 35b40
• 35s15