# Nonexistence of proper *p*-biharmonic maps and Liouville type theorems I: case of \(p\ge 2\)

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## Abstract

Let \(u: (M, g)\rightarrow (N, h)\) be a map between Riemannian manifolds (*M*, *g*) and (*N*, *h*). The *p*-bienergy of *u* is defined by \(E_p(u)=\int _M|\tau (u)|^pd\nu _g\), where \(\tau (u)\) is the tension field of *u* and \(p>1\). Critical points of \(E_p(\cdot )\) are called *p*-biharmonic maps. In this paper we will prove nonexistence result of proper *p*-biharmonic maps when \(p\ge 2\). In particular when \(M=\mathbb {R}^m\), we get Liouville type results under proper integral conditions , which extend the related results of Baird, Fardoun and Ouakkas (2010).

## Keywords

p-biharmonic maps Nonpositive curvature Rigidity## Mathematics Subject Classification

53C24 53C43## Notes

### Acknowledgements

Yingbo Han was supported by NSF of China (No.11971415) and Nanhu Scholars Program for Young Scholars of XYNU and the Universities Young Teachers Program of Henan Province (2016GGJS-096). Yong Luo was supported by the NSF of China (No.11501421). Both authors would like to thank the anonymous reviewer for the suggestions which make this paper more readable.

### Compliance with ethical standards

### Conflict of interest

The authors declare that they have no conflict of interest.

### Human participants or animals

This article does not contain any studies with human participants or animals performed by any of the authors.

### Informed consent

Informed consent was obtained from all individual participants included in the study.

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