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

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} Δ^{\fracα{2}} u &=& u^γ&\ \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.

22 pages