A symmetry problem in the calculus of variations

We consider the integral functional \[ J(u) = \int_{\Omega} [f(|Du|) - u]\, dx\,, \qquad u\in\Wuu(\Omega), \] where $\Omega\subset\R^n$, $n\geq 2$, is a nonempty bounded connected open subset of $\R^n$ with smooth boundary, and $\R\ni s\mapsto f(|s|)$ is a convex, differentiable function. We prove that, if $J$ admits a minimizer in $\Wuu(\Omega)$ depending only on the distance from the boundary of $\Omega$, then $\Omega$ must be a ball