Nonradial singular solutions for elliptic equations with exponential nonlinearity

Jingyue Cao; Yunkang Shao; Fangshu Wan; Jiaqi Wang; Yifei Zhu; Jingyue Cao; Yunkang Shao; Fangshu Wan; Jiaqi Wang; Yifei Zhu

doi:10.3934/era.2024146

Electronic Research Archive

2024, Volume 32, Issue 5: 3171-3201. doi: 10.3934/era.2024146

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Research article

Nonradial singular solutions for elliptic equations with exponential nonlinearity

School of Mathematical Sciences, Anhui University, Hefei 230601, China

Received: 09 January 2024 Revised: 17 April 2024 Accepted: 18 April 2024 Published: 07 May 2024

For any $ R > 0 $, infinitely many nonradial singular solutions can be constructed for the following equation:

$ \begin{equation} -\Delta u = e^u \;\;\; \mbox{in}\; B_R \backslash \{0\} , \;\;\;\;\;\;(0.1)\end{equation} $

where $ B_R = \{x \in \mathbb{R}^N \; (N \geq 3): \; |x| < R\} $. To construct nonradial singular solutions, we need to consider asymptotic expansion at the isolated singular point $ x = 0 $ of a prescribed solution of (0.1). Then, nonradial singular solutions of (0.1) can be constructed by using the asymptotic expansion and introducing suitable weighted Hölder spaces.
- nonradial singular solutions,
- asymptotic expansions,
- exponential nonlinearity,
- weighted spaces
Citation: Jingyue Cao, Yunkang Shao, Fangshu Wan, Jiaqi Wang, Yifei Zhu. Nonradial singular solutions for elliptic equations with exponential nonlinearity[J]. Electronic Research Archive, 2024, 32(5): 3171-3201. doi: 10.3934/era.2024146

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Abstract

For any $ R > 0 $, infinitely many nonradial singular solutions can be constructed for the following equation:

$ \begin{equation} -\Delta u = e^u \;\;\; \mbox{in}\; B_R \backslash \{0\} , \;\;\;\;\;\;(0.1)\end{equation} $

where $ B_R = \{x \in \mathbb{R}^N \; (N \geq 3): \; |x| < R\} $. To construct nonradial singular solutions, we need to consider asymptotic expansion at the isolated singular point $ x = 0 $ of a prescribed solution of (0.1). Then, nonradial singular solutions of (0.1) can be constructed by using the asymptotic expansion and introducing suitable weighted Hölder spaces.

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