A viscosity equation for minimizers of a class of very degenerate elliptic functionals

We consider the functional $$J(v) = \int_\Omega [f(|\nabla v|) - v] dx,$$ where $\Omega$ is a bounded domain and $f:[0,+\infty)\to \mathbb{R}$ is a convex function vanishing for $s\in [0,\sigma]$, with $\sigma>0$. We prove that a minimizer $u$ of $J$ satisfies an equation of the form $$\min(F(\nabla u, D^2 u), |\nabla u|-\sigma)=0$$ in the viscosity sense.