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

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.