Singular solutions for divergence-form elliptic equations involving regular variation theory: Existence and classification

TL;DR

This paper classifies isolated singularities of nonlinear elliptic equations with divergence form, establishing sharp conditions for removability and describing asymptotic behaviors of solutions near singularities.

Contribution

It extends previous results by providing a complete classification of singularities using regular variation theory and introduces new techniques for analyzing asymptotic behaviors.

Findings

Sharp condition for singularity removability: $b(x)h(\

classification of singularities as weak or strong based on asymptotic limits

Development of new methods for analyzing asymptotic behavior in critical cases

Abstract

We generalise and sharpen several recent results in the literature regarding the existence and complete classification of the isolated singularities for a broad class of nonlinear elliptic equations of the form \begin{equation} -{\rm div}\,(\mathcal A(|x|) \,|\nabla u|^{p-2} \nabla u)+b(x)\,h(u)=0\quad \text{in } B_1\setminus\{0\}, \end{equation} where $B_r$ denotes the open ball with radius $r>0$ centred at zero in $\mathbb{R}^N$ $(N\geq 2)$. We assume that $\mathcal{A} \in C^1(0,1]$, $b\in C(\bar{B_1}\setminus\{0\})$ and $h\in C[0,\infty)$ are positive functions associated with regularly varying functions of index $\vartheta$, $\sigma$ and $q$ at $0$, $0$ and $\infty$ respectively, satisfying $q>p-1>0$ and $\vartheta-\sigma<p<N+\vartheta$. We prove that the condition $b(x) \,h(\Phi)\not \in L^1(B_{1/2})$ is sharp for the removability of all singularities at zero for the positive solutions of our problem, where $\Phi$ denotes the "fundamental solution" of $-{\rm div}\,(\mathcal A(|x|)\, |\nabla u|^{p-2} \nabla u)=\delta_0$ (the Dirac mass at zero) in $B_1$, subject to $\Phi|_{\partial B_1}=0$. If $b(x) \,h(\Phi)\in L^1(B_{1/2})$, we show that any non-removable singularity at zero for a positive solution to our equation is either weak (i.e., $\lim_{|x|\to 0} u(x)/\Phi(|x|)\in (0,\infty)$) or strong ($ \lim_{|x|\to 0} u(x)/\Phi(|x|)=\infty$). The main difficulty and novelty of this paper, for which we develop new techniques, come from the explicit asymptotic behaviour of the strong singularity solutions in the critical case, which had previously remained open even for $\mathcal{A}=1$. We also study the existence and uniqueness of the positive solution to our problem with a prescribed admissible behaviour at zero and a Dirichlet condition on $\partial B_1$.