Blow up boundary solutions of some semilinear fractional equations in the unit ball

TL;DR

This paper investigates boundary blow-up solutions for a fractional semilinear PDE in the unit ball, highlighting different singularity behaviors and demonstrating that classical conditions do not apply in the fractional context.

Contribution

It characterizes boundary blow-up solutions for fractional equations, distinguishing singularity types, and shows the classical Keller-Osserman condition is not suitable for fractional cases.

Findings

Identified two types of boundary singularities for solutions.

Proved classical Keller-Osserman condition does not hold in fractional setting.

Analyzed the asymptotic behavior of solutions near the boundary.

Abstract

For $\gamma>0$, we are interested in blow up solutions $u\in C^+(B)$ of the fractional problem in the unit ball $B$ \begin{equation}\label{2nov} \left\{\begin{array} {rcll} \Delta^{\frac{\alpha}{2}} u &=& u^\gamma&\ \text{in }B\\ u &=& 0&\ \text{in }B^c.\end{array}\right. \end{equation} We distinguish particularly two orders of singularity at the boundary: solutions exploding at the same rate than $\delta^{1-\frac{\alpha}{2}}$ ($\delta$ denotes the Euclidean distance) and those higher singular than $\delta^{1-\frac{\alpha}{2}}.$ As a consequence, it will be shown that the classical Keller-Osserman condition can not be readopted in the fractional setting.