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

TL;DR

This paper investigates the existence of large solutions to fractional Laplacian equations with boundary singularities, extending classical concepts to the fractional setting and establishing conditions for their existence.

Contribution

It introduces a fractional Keller-Osserman condition and provides sufficient criteria for solutions with boundary singularities in fractional Laplacian problems.

Findings

Established existence conditions for large solutions with boundary singularities.

Extended classical large solution concepts to fractional Laplacian equations.

Identified fractional analogs of Keller-Osserman conditions.

Abstract

We look for solutions of $(-\Delta)^s u+f(u) = 0$ in a bounded smooth domain $\Omega$, $s\in(0,1)$, with a strong singularity at the boundary. In particular, we are interested in solutions which are $L^1(\Omega)$ and higher order with respect to 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.