Abstract We present the sharp characterization of the behavior at the isolated singularities of positive solutions of some equations on singular manifolds with conical metrics. It is seen that the equations on singular manifolds with conical metrics are equivalent to weighted elliptic equations in where is the unit ball. The weights can be singular at x = 0. We present the sharp asymptotic behavior of positive solutions of the weighted elliptic equations at x = 0 and establish expansions of these solutions up to arbitrary orders. Asymptotic behavior at the isolated singularitie of positive solutions of elliptic equations without weights has been studied by many authors. We will obtain new results on the asymptotic behavior at the isolated singularities even for positive solutions of equations without weights in the subcritical case.

中文翻译：

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锥度规奇异流形上某些方程解的孤立奇点处的渐近行为

摘要 我们提出了在具有圆锥度量的奇异流形上的一些方程的正解的孤立奇点处行为的清晰表征。可见，具有圆锥度量的奇异流形方程等价于加权椭圆方程，其中 是单位球。权重在 x = 0 处可以是奇异的。我们在 x = 0 处呈现加权椭圆方程的正解的尖锐渐近行为，并建立这些解的扩展到任意阶数。许多作者已经研究了无权椭圆方程的正解的孤立奇点处的渐近行为。即使对于亚临界情况下没有权重的方程的正解，我们也将获得关于孤立奇点处渐近行为的新结果。