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

We consider the functional $$J(v)={\int }_{\mathrm{\Omega }}[f(\mathrm{\mid }\mathrm{\nabla }v\mathrm{\mid })-v]dx,$$ where $\mathrm{\Omega }$ is a bounded domain and $f:[0,+\mathrm{\infty })\to \mathbb{R}$ 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 $$\min (F(\mathrm{\nabla }u,D^{2}u),\mathrm{\mid }\mathrm{\nabla }u\mathrm{\mid }-\sigma )=0$$ in the viscosity sense.