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

where $B_r$ denotes the open ball with radius $r>0$ centred at $0$ in ${\mathbb{R}}^N$ $(N\ge 2)$. We assume that $\mathcal{A}\in C^1(0,1]$, $b\in C(\overline{B_1}\setminus \{0\})$ 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 $b(x)h(\mathrm{\Phi })\notin L^1(B_{1/2})$ is sharp for the removability of all singularities at $0$ for the positive solutions of $(0.1)$, where $\Phi$ denotes the “fundamental solution” of $-\mathrm{div}(\mathcal{A}(|x|)|\nabla u{|}^{p-2}\nabla u)={\delta }_0$ (the Dirac mass at 0) in $B_1$, subject to $\mathrm{\Phi }{|}_{\partial B_1}=0$. If $b(x)h(\mathrm{\Phi })\in L^1(B_{1/2})$, we show that any non-removable singularity at $0$ for a positive solution of (0.1) is either weak (i.e., ${\lim }_{|x|\to 0}u(x)/\mathrm{\Phi }(|x|)\in (0,\infty )$) or strong (${\lim }_{|x|\to 0}u(x)/\mathrm{\Phi }(|x|)=\infty$). 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 of $(0.1)$ with a prescribed admissible behaviour at $0$ and a Dirichlet condition on $\partial B_1$.