Applications of the $s$-harmonic extension to the study of singularities of Emden's equations

We use the Caffarelli–Silvestre extension to $\mathbb{R}_+\times\mathbb{R}^N$ to study the isolated singularities of functions satisfying the semilinear fractional equation $(-\Delta)^sv+\epsilon v^p=0$ in a punctured domain of $\mathbb{R}^N$ where $\epsilon=\pm 1$, $0<s<1$ and $p>1$. We emphasise the obtention of a priori estimates and analyse the set of self-similar solutions. We provide a complete description of the possible behaviour of solutions near a singularity.