We study asymptotic behavior of singular solutions of \(\Delta u+e^u=0\) in \(B \backslash \{0\}\), where \(B=\{x \in {\mathbb {R}}^2: \; |x|<1\}\). It is known that if \(u \in C^2(B \backslash \{0\})\) with \(\int _{B \backslash \{0\}} e^u dx<\infty \) is a singular solution to the equation, there is \(\alpha _0>-2\) such that

$$\begin{aligned} u(x)=\alpha _0 \ln |x|+O(1) \;\; \text{ as } |x| \rightarrow 0. \end{aligned}$$

We will establish asymptotic expansions up to arbitrary orders of u(x) near the isolated singular point \(x=0\). This provides the “sharp” results for the asymptotic behavior of u(x) in some sense.

中文翻译：

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穿孔盘中$$ \ Delta u + e ^ u = 0 $$Δu + eu = 0的奇异解的渐近展开

我们研究的单数解的渐近行为\（\德尔塔U + E 1 U = 0 \）在\（B \反斜杠\ {0 \} \） ，其中\（B = \ {X \在{\ mathbb {R }} ^ 2：\; | x | <1 \} \）。众所周知，如果\（u \ in C ^ 2（B \反斜杠\ {0 \}）\）与\（\ int _ {B \反斜杠\ {0 \}} e ^ u dx <\ infty \）是方程的奇异解，存在\（\ alpha _0> -2 \）这样

$$ \ begin {aligned} u（x）= \ alpha _0 \ ln | x | + O（1）\; \; \ text {as} | x | \ rightarrow0。\ end {aligned} $$

我们将在孤立的奇异点\（x = 0 \）附近建立直至u（x）任意阶的渐近展开。在某种意义上，这为u（x）的渐近行为提供了“清晰”的结果。