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 $-b^2\le K\le -a^2<0$, $b\ge a>0.$ 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 $$\mathcal{Q}(u)=\mathrm{div}\left(\frac{\mathcal{A}(|\nabla u|)}{|\nabla u|}\nabla u\right)=0,$$ 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.

References in corpus (2)

• Asymptotic Dirichlet problem for A-harmonic and minimal graph equations in Cartan-Hadamard manifolds
• A note on minimal graphs over certain unbounded domains of Hadamard manifolds

Cited by in corpus (1)

• On the asymptotic Dirichlet problem for a class of mean curvature type partial differential equations