Self-similar solutions of energy-supercritical focusing wave equations in all dimensions

In this paper, we prove the existence of a countable family of regular spherically symmetric self-similar solutions to focusing energy super-critical semi-linear wave equations \begin{equation*} \partial_{tt}u-\Delta u=|u|^{p-1}u \qquad $\text{in}$ ,, $\mathbb{R}$^{N}, \end{equation*} where $N\ge 3$, $1+\frac{4}{N-2}<p$, and, if $N\ge 4$, $p \leq 1+\frac{4}{N-3}$\mathrm{.}Thiswaspreviouslyknownonlyinthecase$N=3$, for integer $p$ (see Bizo'{n}, Maison and Wasserman \cite{BMW}). We also study the asymptotics of these solutions.