On a class of rotationally symmetric $p$-harmonic maps

L. F. Cheung; C. K. Law; M. C. Leung

doi:10.3934/cpaa.2017095

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2017, Volume 16, Issue 6: 1941-1955. Doi: 10.3934/cpaa.2017095

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On a class of rotationally symmetric $p$-harmonic maps

1.
Department of Mathematics, The Chinese University of Hong Kong, Shatin, Hong Kong
2.
Department of Applied Mathematics, National Sun Yat-sen University, Kaohsiung, Taiwan 80424, Taiwan
3.
Department of Mathematics, National University of Singapore, Singapore 119260, Singapore

Received: September 30, 2014

Revised: February 28, 2017

Published: September 2017

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Abstract

We give a classification of rotationally symmetric $p$-harmonic maps between some model spaces such as $\mathbb{R}^n$ and $\mathbb{H}^n$ by their asymptotic behaviors. Among others, we show that, when $p>2$ and $n≥q 2$, all rotationally symmetric $p$-harmonic maps from $\mathbb{R}^n$ to $\mathbb{H}^n$ have to blow up at a finite point, while all rotationally symmetric $p$-harmonic maps from $\mathbb{H}^n$ to $\mathbb{H}^n$ observe the trichotomy property, i.e. the map $y$ is the identity map, is bounded or blows up according as its initial value $y'(0)$ is equal to, less than or greater than one. Our sharp estimates imply and improve a number of existence and non-existence results of certain $p$-harmonic maps on noncompact manifolds.

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Mathematics Subject Classification: 34C13, 58E20, 58E.

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