### 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 |
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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 |

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### 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