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Communications in Analysis and Geometry
Volume 27 (2019)
Number 4
On the nature of isolated asymptotic singularities of solutions of a family of quasi-linear elliptic PDE's on a Cartan–Hadamard manifold
Pages: 791 – 807
DOI: https://dx.doi.org/10.4310/CAG.2019.v27.n4.a2
Authors
Abstract
Let $M$ be a Cartan–Hadamard manifold with sectional curvature satisfying $-b^2 \leq K \leq -a^2 \lt 0, b \geq a \gt 0$. 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 $p_1, \dotsc , p_k \in \partial_{\infty} M$, if $u \in C^1(M) \cap C^0 ( \bar{M} \setminus \lbrace p_1, \dotsc , p_k \rbrace )$ satisfies a PDE $\mathcal{Q}(u) = 0$ in $M$ and if $u \vert {}_{\partial_{\infty} M \setminus \lbrace p_1, \dotsc , p_k \rbrace}$ extends continuously to $p_i, i = 1, \dotsc , k$, can one conclude that $u \in C^0 \bar{M}$? When $\mathrm{dim} \: M = 2$, for $\mathcal{Q}$ belonging to a linearly convex space of quasilinear elliptic operators $\mathcal{S}$ of the form\[\mathcal{Q}(u) = \mathrm{div} \: \bigl(\frac {( \mathcal{A} ( \lvert \nabla u \rvert ) }{ \lvert \nabla u \rvert } \nabla u\bigr) = 0 \quad \textrm{,}\]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 $\mathbb{H}^n , n \geq 2$, 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, $p \gt 1$, 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.
Received 25 May 2016
Accepted 23 March 2017
Published 8 October 2019