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

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 $\partial B_1$.