Self-similar solutions of focusing semi-linear wave equations in $${\mathbb {R}}^{N}$$

In this paper, we prove the existence of a countable family of regular spherically symmetric self-similar solutions to focusing energy-supercritical semi-linear wave equations $$\begin{aligned} \partial _{tt}u-\Delta u=|u|^{p-1}u \qquad \text {in} \,\, \mathbb {R}^{N}, \end{aligned}$$ where $$N\ge 3$$ , $$1+\frac{4}{N-2}<p$$ and, if $$N\ge 4$$ , $$p \le 1+\frac{4}{N-3}$$ . This was previously known only in the case $$N=3$$ , for integer p (see Bizon et al. in Nonlinearity 20(9):2061–2074, 2007). We also study the asymptotics of these solutions.