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Liouville properties for $ p$-harmonic maps with finite $ q$-energy


Authors: Shu-Cheng Chang, Jui-Tang Chen and Shihshu Walter Wei
Journal: Trans. Amer. Math. Soc. 368 (2016), 787-825
MSC (2010): Primary 53C21, 53C24; Secondary 58E20, 31C45
DOI: https://doi.org/10.1090/tran/6351
Published electronically: September 9, 2015
MathSciNet review: 3430350
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Abstract: We introduce and study an approximate solution of the $ p$-Laplace equation and a linearlization $ \mathcal {L}_{\epsilon }$ of a perturbed $ p$-Laplace operator. By deriving an $ \mathcal {L}_{\epsilon }$-type Bochner's formula and Kato type inequalities, we prove a Liouville type theorem for weakly $ p$-harmonic functions with finite $ p$-energy on a complete noncompact manifold $ M$ which supports a weighted Poincaré inequality and satisfies a curvature assumption. This nonexistence result, when combined with an existence theorem, yields in turn some information on topology, i.e. such an $ M$ has at most one $ p$-hyperbolic end. Moreover, we prove a Liouville type theorem for strongly $ p$-harmonic functions with finite $ q$-energy on Riemannian manifolds. As an application, we extend this theorem to some $ p$-harmonic maps such as $ p$-harmonic morphisms and conformal maps between Riemannian manifolds. In particular, we obtain a Picard-type theorem for $ p$-harmonic morphisms.


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Additional Information

Shu-Cheng Chang
Affiliation: Department of Mathematics, National Taiwan University, Taipei 10617, Taiwan, Republic of China
Email: scchang@math.ntu.edu.tw

Jui-Tang Chen
Affiliation: Department of Mathematics, National Taiwan Normal University, Taipei 11677, Taiwan, Republic of China
Email: jtchen@ntnu.edu.tw

Shihshu Walter Wei
Affiliation: Department of Mathematics, University of Oklahoma, Norman, Oklahoma 73019-0315
Email: wwei@ou.edu

DOI: https://doi.org/10.1090/tran/6351
Keywords: $p$-harmonic map, weakly $p$-harmonic function, perturbed $p$-Laplace operator, $p$-hyperbolic end, Liouville type properties
Received by editor(s): November 13, 2012
Received by editor(s) in revised form: October 30, 2013, and December 3, 2013
Published electronically: September 9, 2015
Additional Notes: The first and second authors were partially supported by the NSC
The third author was partially supported by the NSF(DMS-1447008) and the OU Arts and Sciences Travel Assistance Program Fund
Article copyright: © Copyright 2015 American Mathematical Society

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