  In this paper, we investigate the minimality of the map $\frac{x}{\|x\|}$
from the euclidean unit ball $\mathbf{B}^n$ to its boundary $\mathbb{S}^{n-1}$
for weighted energy functionals of the type $E\_{p,f}=
\int\_{\mathbf{B}^n}f(r)\|\nabla u\|^p dx$, where $f$ is a non-negative
function. We prove that in each of the two following cases: i) $p=1$ and $f$ is
non-decreasing, i)) $p$ is an integer, $p \leq n-1$ and $f= r^{\alpha}$ with
$\alpha \geq 0$, the map $\frac{x}{\|x\|}$ minimizes $E\_{p,f}$ among the maps
in $W^{1,p}(\mathbf{B}^n, \mathbb{S}^{n-1})$ which coincide with
$\frac{x}{\|x\|}$ on $\partial \mathbf{B}^n$. We also study the case where $
f(r)= r^{\alpha}$ with $-n+2 < \alpha < 0$ and prove that $\frac{x}{\|x\|}$
does not minimize $E\_{p,f}$ for $\alpha$ close to $-n+2$ and when $n \geq 6$,
for $\alpha$ close to $4-n$.
