### Abstract

We introduce and study an approximate solution of the p-Laplace equation and a linearlization ℒ_{ϵ} of a perturbed p-Laplace operator. By deriving an ℒ_{ϵ}-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.

Original language | English |
---|---|

Pages (from-to) | 787-825 |

Number of pages | 39 |

Journal | Transactions of the American Mathematical Society |

Volume | 368 |

Issue number | 2 |

DOIs | |

Publication status | Published - 2016 Feb |

### Fingerprint

### Keywords

- Liouville type properties
- Perturbed p-Laplace operator
- Weakly p-harmonic function
- p-harmonic map
- p-hyperbolic end

### ASJC Scopus subject areas

- Mathematics(all)
- Applied Mathematics

### Cite this

*Transactions of the American Mathematical Society*,

*368*(2), 787-825. https://doi.org/10.1090/tran/6351

**Liouville properties for p-harmonic maps with finite q-energy.** / Chang, Shu Cheng; Chen, Jui Tang; Wei, Shihshu Walter.

Research output: Contribution to journal › Article

*Transactions of the American Mathematical Society*, vol. 368, no. 2, pp. 787-825. https://doi.org/10.1090/tran/6351

}

TY - JOUR

T1 - Liouville properties for p-harmonic maps with finite q-energy

AU - Chang, Shu Cheng

AU - Chen, Jui Tang

AU - Wei, Shihshu Walter

PY - 2016/2

Y1 - 2016/2

N2 - We introduce and study an approximate solution of the p-Laplace equation and a linearlization ℒϵ of a perturbed p-Laplace operator. By deriving an ℒϵ-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.

AB - We introduce and study an approximate solution of the p-Laplace equation and a linearlization ℒϵ of a perturbed p-Laplace operator. By deriving an ℒϵ-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.

KW - Liouville type properties

KW - Perturbed p-Laplace operator

KW - Weakly p-harmonic function

KW - p-harmonic map

KW - p-hyperbolic end

UR - http://www.scopus.com/inward/record.url?scp=84951937520&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84951937520&partnerID=8YFLogxK

U2 - 10.1090/tran/6351

DO - 10.1090/tran/6351

M3 - Article

AN - SCOPUS:84951937520

VL - 368

SP - 787

EP - 825

JO - Transactions of the American Mathematical Society

JF - Transactions of the American Mathematical Society

SN - 0002-9947

IS - 2

ER -