Lipschitz regularity for local minimizers of some widely degenerate problems

We consider local minimizers of the functional $$\sum _{i=1}^N\int (\mathrm{\mid }{u}_{x_i}\mathrm{\mid }-{\delta }_i{)}_{+}^{p}dx+\int fudx,$$ where ${\delta }_1,\dots ,{\delta }_N\ge 0$ and $(\cdot {)}_{+}$ stands for the positive part. Under suitable assumptions on $f$, we prove that local minimizers are Lipschitz continuous functions if $N=2$ and $p\ge 2$, or if $N\ge 2$ and $p\ge 4$.