Weak comparison principle for widely degenerate elliptic equations

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