Assume $\mathrm{\Omega }\subset {ℝ}^2$ is a planar domain, and $u$ is a locally bounded distributional solution to the elliptic equation $$-\mathrm{\Delta }u=|x{|}^{2\beta }h(x)f(u)\text{in }\mathrm{\Omega },$$ where $\beta >-1$ is a constant, $h$ and $f$ are real analytic functions defined on $\mathrm{\Omega }$ and the real line $ℝ$, respectively. We establish asymptotic expansions of $u(x)$ to arbitrary orders near $0$, which complements the recent results of Han–Li–Li on the Yamabe equation, Guo–Li–Wanon the weighted Yamabe equation, and partly extends that of Guo–Wan–Yang on the Liouville equation in a punctured disc. Our method is a combination of a priori estimate and mathematical induction.

中文翻译：

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椭圆方程二维分布解的展开

认为$\mathrm{\Omega }\subset {ℝ}^2$是一个平面域，并且$你$是椭圆方程的局部有界分布解$$-\mathrm{\Delta }你=|X{|}^{2\beta }H(X)F(你)\text{在 }\mathrm{\Omega },$$在哪里$\beta >-1$是一个常数，$H$和$F$是定义在上的实分析函数$\mathrm{\Omega }$和实线$ℝ$， 分别。我们建立的渐近展开$你(X)$到附近的任意订单$0$，它补充了 Han-Li-Li 在 Yamabe 方程上的最新结果，Guo-Li-Wanon 是加权 Yamabe 方程，并部分扩展了 Guo-Wan-Yang 在穿孔圆盘中的 Liouville 方程上的结果。我们的方法是先验估计和数学归纳的结合。