Nonradial singular solutions for elliptic equations with exponential nonlinearity

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.

Keywords

• nonradial singular solutions
• asymptotic expansions
• exponential nonlinearity
• weighted spaces