| Abstract: |
| We study the existence and complete classification of the isolated singularities for weighted divergence-form equations of the form $${\rm div} (\mathcal{A}(|x|) |\nabla u|^{p-2}\nabla u)=b(x)h(u) \quad \text{in } B_1 (0) \setminus \{0\}.$$ We assume that $\mathcal{A} \in C^1 (0,1]$, $b\in C(B_1\setminus \{0\})$ and $h\in C[0,\infty)$ are positive functions associated with regularly varying functions of index $\theta$, $\sigma$, and $q$ at $0$, $0$, and $\infty$ respectively.
We reveal how the interplay between these indices affect the classification of the singular solutions near the singularity. We are particularly interested in the so-called critical cases of the indices which are important in the non-power nonlinearity case as they represent the threshold between having a trichotomous classification, a dichotomous classification or no singularities at all. Our results complement a series of works on removable and non-removable singularities in the framework of pure-power nonlinearities, where the difficulties that arise from regular variation do not appear. |
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