  In this paper, we investigate minimizing properties of the map $x/\|x\|$ from
the Euclidean unit ball $\mathbf{B}^{n}$ to its boundary $\mathbb{S}^{n-1}$,
for the weighted energy functionals $E^n\_{p,\alpha}(u)=\int\_{\mathbf{B}^{n}}
\|x\|^{\alpha}\|\nabla u\|^p dx$. We establish the following induction
principle: if the map $\frac{x}{\|x\|}:\mathbf{B}^{n+1}\to \mathbb{S}^n$
minimizes $E^{n+1}\_{p,\alpha}$ among the maps $u: \mathbf{B}^{n+1}\to
\mathbb{S}^n$ satisfying $u(x)=x$ on $\mathbb{S}^n$, then the map
$\frac{y}{\|y\|}:\mathbf{B}^n\to\mathbb{S}^{n-1}$ minimizes
$E^{n}\_{p,\alpha+1}$ among the maps $v: \mathbf{B}^n\to\mathbb{S}^{n-1}$
satisfying $v(y)=y$ on $\mathbb{S}^{n-1}$. This result enables us to enlarge
the range of values of $p$ and $\alpha$ for which $x/\|x\|$ minimizes
$E^n\_{p,\alpha}$.
