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 \(\begin{aligned} -\text {div} \left( \left( |Du|-1 \right) ^{p-1}_+\frac{Du}{|Du|} \right) = f(x,u) \qquad \text { in } \Omega , \end{aligned}\) $$\begin{array}{c}\hfill -\text{div}\left({\left(|,D,u,|,-,1\right)}_{+}^{p-1},\frac{\mathit{Du}}{|Du|}\right)=f(x,u)\text{in}\mathrm{\Omega },\end{array}$$ where \(p \ge 2\) $p\ge 2$ and \(\Omega \) $\mathrm{\Omega }$ is an open subset of \(\mathbb {R}^{n}\) ${\mathbb{R}}^n$ , \(n\ge 2\) $n\ge 2$ . Moreover, we establish some second order regularity results of the solutions, that yields a weighted Sobolev inequality with widely degenerate weights.