Nonexistence of proper $p$-biharmonic maps and Liouville type theorems I: case of $p\ge 2$

Abstract

Let $u:(M,g)\to (N,h)$ be a map between Riemannian manifolds $(M,g)$ and $(N,h)$. The $p$-bienergy of $u$ is defined by $E_p(u)={\int }_M|\tau (u){|}^{p}d{\nu }_g$, 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 \cite{BFO}.

arXiv commentTitle changed; to appear in Journal of Elliptic and Parabolic Equations

References in corpus (1)

• A short survey on biharmonic maps between Riemannian manifolds