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

TL;DR

This paper investigates the boundary behavior of solutions to certain quasi-linear elliptic PDEs on Cartan-Hadamard manifolds, establishing conditions under which solutions extend continuously to the boundary at infinity, with complete results in hyperbolic space.

Contribution

It characterizes when solutions with boundary data excluding finitely many points extend continuously to the boundary at infinity for a broad class of quasi-linear elliptic PDEs, especially in hyperbolic space.

Findings

Solutions extend continuously to the boundary at infinity under certain growth conditions.

The class of PDEs splits into two types: those with positive and negative boundary extension results.

Explicit solutions with isolated non-removable singularities at infinity are constructed for the negative case.

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.