Infinitely many nonradial singular solutions of $\Delta u+e^u=0$ in $\mathbb{R}^N\backslash\{0\}$, $4\le N\le 10$

Abstract

We construct countably infinitely many nonradial singular solutions of the problem \[ \Delta u+e^u=0\ \ \textrm{in}\ \ \mathbb{R}^N\backslash\{0\},\ \ 4\le N\le 10 \] of the form \[ u(r,\sigma)=-2\log r+\log 2(N-2)+v(\sigma), \] where $v(\sigma)$ depends only on $\sigma\in \mathbb{S}^{N-1}$. To this end we construct countably infinitely many solutions of \[ \Delta_{\mathbb{S}^{N-1}}v+2(N-2)(e^v-1)=0,\ \ 4\le N\le 10, \] using ODE techniques.