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arXiv · 1501.03878

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

Yasuhito Miyamoto
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Abstract

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

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Yasuhito Miyamoto. 2019-12-24. Infinitely many nonradial singular solutions of $Δu+e^u=0$ in $\mathbb{R}^N\backslash\{0\}$, $4\le N\le 10$. https://doi.org/10.1017/s0308210517000051

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