Weakly biharmonic maps from the ball to the sphere

Abstract

The aim of this paper is to investigate the existence of proper, weakly biharmonic maps within a family of rotationally symmetric maps $$u_a : B^n \rightarrow {\mathbb {S}}^n$$, where $$B^n$$ and $${\mathbb {S}}^n$$ denote the Euclidean n-dimensional unit ball and sphere respectively. We prove that there exists a proper, weakly biharmonic map $$u_a$$ of this type if and only if $$n=5$$ or $$n=6$$. We shall also prove that these critical points are unstable.