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

TL;DR

This paper constructs infinitely many nonradial singular solutions to a nonlinear PDE in rom or dimensions 4 to 10, revealing complex solution structures using spherical and ODE methods.

Contribution

It introduces a novel construction of infinitely many nonradial singular solutions for a specific PDE in certain dimensions, expanding understanding of solution diversity.

Findings

Existence of countably infinite nonradial singular solutions.

Solutions are constructed via spherical PDE and ODE techniques.

Results apply to dimensions 4 through 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.