Biharmonic maps are critical point of the bienergy functional \[ E_2(u)=\frac{1}{2}\int_M|\tau(u)|^2 d\,v_g, \] where \(\tau(u)=\textrm{trace}\,\nabla d u\). In particular, harmonic maps are trivially biharmonic maps, as harmonic maps are critical points of the energy functional \[ E(u)=\frac{1}{2}\int_M| d u |^2 d\,v_g. \] Biharmonic maps that are not harmonic are called proper biharmonic maps.
Let \(B^n\) and \(S^n\) denote the \(n\)-dimensional Euclidean unit ball and sphere respectively. The authors study the family of rotationally symmetric maps \(u_a:B^n\rightarrow S^n\subset \mathbb{R}^n\times \mathbb{R}\) given by \[ u_a(x)=\left(\sin a \,\,\frac{x}{|x|},\cos a\right), \] where \(a\) is a constant in \((0,\pi/2)\). In particular, the authors show (Theorem 1.1) that such maps are proper weakly biharmonic if and only if if either \( n = 5\) and \(a = \pi/3\) or \(n = 6\) and \(a = 1/2 \arccos(-4/5)\). In any of these two cases, \(u_a\) is unstable (Theorem 1.2).
The paper has three sections and it is quite self-contained, with a recollection of the necessary results on Sobolev spaces and weak solutions presented in section 2. Proofs of the main results are given in section 3.