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# Communications in Analysis and Geometry

## Volume 27 (2019)

### Number 4

### Asymptotic Dirichlet problem for ${\mathcal A}$-harmonic and minimal graph equations in Cartan–Hadamard manifolds

Pages: 809 – 855

DOI: https://dx.doi.org/10.4310/CAG.2019.v27.n4.a3

#### Authors

#### Abstract

We study the asymptotic Dirichlet problem for $\mathcal{A}$-harmonic equations and for the minimal graph equation on a Cartan–Hadamard manifold $M$ whose sectional curvatures are bounded from below and above by certain functions depending on the distance $r =d(\cdot , o)$ to a fixed point $o \in M$. We are, in particular, interested in finding optimal (or close to optimal) curvature upper bounds. In the special case of the Laplace–Beltrami equation we are able to solve the asymptotic Dirichlet problem in dimensions $n \geq 3$ if radial sectional curvatures satisfy\[\dfrac {(\log (r(x))^{2 \overline{\varepsilon}}}{r(x)^2} \leq K \leq - \dfrac {1 + \varepsilon}{r(x)^2 \log r(x)}\]outside a compact set for some $\varepsilon \gt \overline{\varepsilon} \gt 0$. The upper bound is closeto optimal since the nonsolvability is known if\[K \geq -1 / (2r(x)^2 \log r(x)) \quad \textrm{.}\]Our results (in the non-rotationally symmetric case) improve on the previously known case of the quadratically decaying upper bound.

J.-B.C. is supported by the CNPq (Brazil) project 501559/2012-4. I.H. is supported by the Academy of Finland, project 252293; J.R. supported by the CNPq (Brazil) project 302955/2011-9.

Received 6 March 2016

Accepted 24 April 2017

Published 8 October 2019