On the nature of isolated asymptotic singularities of solutions of a family of quasi-linear elliptic PDE's on a Cartan–Hadamard manifold

Abstract

Let $M$ be a Cartan-Hadamard manifold with sectional curvature satisfying $-b^2\leq K\leq -a^2<0$, $b\geq a>0.$ Denote by $\partial_{\infty}M$ the asymptotic boundary of $M$ and by $\bar M:= M\cup\partial_\infty M$ the geometric compactification of $M$ with the cone topology. We investigate here the following question: Given a finite number of points $p_{1},...,p_{k}\in\partial_\infty M,$ if $u\in C^{\infty}(M)\cap C^{0}\left( \bar{M}\backslash\left\{ p_{1},...,p_{k}\right\} \right) $ satisfies a PDE $\mathcal Q(u)=0$ in $M$ and if $u|_{\partial_\infty M\setminus\left\{p_{1},...,p_{k}\right\}}$ extends continuously to $p_{i},$ $i=1,...,k,$ can one conclude that $u\in C^{0}\left( \bar{M}\right )?$ When $\dim M=2$, for $\mathcal Q$ belonging to a linearly convex space of quasi-linear elliptic operators $\mathcal{S}$ of the form $$ \mathcal{Q}(u)=\operatorname{div}\left( \frac{\mathcal{A}(|\nabla u|)}{|\nabla u|} \nabla u \right)=0, $$ where $\mathcal{A}$ satisfies some structural conditions, then the answer is yes provided that $\mathcal{A}$ has a certain asymptotic growth. This condition includes, besides the minimal graph PDE, a class of minimal type PDEs. In the hyperbolic space $ \mathbb{H}^n$, $n\ge 2,$ we are able to give a complete answer: we prove that $\mathcal{S}$ splits into two disjoint classes of minimal type and $p-$Laplacian type PDEs, $p>1,$ where the answer is yes and no respectively. These two classes are determined by the asymptotic behaviour of $\mathcal A.$ Regarding the class where the answer is negative, we obtain explicit solutions having an isolated non removable singularity at infinity.