Liouville properties for $p$-harmonic maps with finite $q$-energy
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- by Shu-Cheng Chang, Jui-Tang Chen and Shihshu Walter Wei PDF
- Trans. Amer. Math. Soc. 368 (2016), 787-825 Request permission
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.References
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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
- MR Author ID: 197212
- Email: wwei@ou.edu
- 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 - © Copyright 2015 American Mathematical Society
- 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
- MathSciNet review: 3430350