Weak comparison principle for widely degenerate elliptic equations

We prove a comparison principle for local weak solutions to a class of widely degenerate elliptic equations of the form \begin{equation} -\text{div} \left( \left(|Du|-1 \right)^{p-1}_+\frac{Du}{|Du|} \right) = f(x,u) \qquad \text{ in } \Omega,\notag \end{equation} where $p \ge 2$ and $\Omega$ is an open subset of $\mathbb{R}^{n}$, $n\geq2$. Moreover, we establish some second order regularity results of the solutions, that yields a weighted Sobolev inequality with widely degenerate weights.