We consider the structure of radially symmetric singular solutions for elliptic equations with the Hardy term and power nonlinearity. In the critical case, it is shown that there exists a unique non-oscillatory singular solution, around which infinitely many singular solutions are oscillating. We also study the subcritical and supercritical cases and make clear the difference of structure from the critical case. Our results can be applied to various problems such as the minimizing problem related to the Caffarelli–Kohn–Nirenberg inequality, the scalar field equation and a self-replication model.

中文翻译：

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带势非线性椭圆方程奇异解的定性分析

我们考虑具有哈代项和幂非线性的椭圆方程的径向对称奇异解的结构。在临界情况下，证明存在唯一的非振荡奇异解，无穷多个奇异解围绕它振荡。我们还研究了亚临界和超临界情况，并明确了结构与临界情况的区别。我们的结果可以应用于各种问题，例如与 Caffarelli-Kohn-Nirenberg 不等式相关的最小化问题、标量场方程和自复制模型。