  We study existence and regularity of weak solutions for the following PDE $$
-\dive(A(x,u)|\nabla u|^{p-2}\nabla u) = f(x,u),\;\;\mbox{in $B_1$}. $$ where
$A(x,s) = A_+(x)\chi_{\{s>0\}}+A_-(x)\chi_{\{s\le 0\}}$ and $f(x,s) =
f_+(x)\chi_{\{s>0\}}+f_-(x)\chi_{\{s\le 0\}}$. Under the ellipticity assumption
that $\frac{1}{\mu}\le A_{\pm} \le \mu$, $A_{\pm}\in C(\O)$ and $f_{\pm}\in
L^N(\O)$, we prove that under appropriate conditions the PDE above admits a
weak solution in $W^{1,p}(B_1)$ which is also $C^{0,\alpha}_{loc}$ for every
$\alpha\in (0,1)$ with precise estimates. Our methods relies on similar
techniques as those developed by Caffarelli to treat viscosity solutions for
fully non-linear PDEs (c.f. \cite{C89}). Other key ingredients in our proofs
are the $\TT_{a,b}$ operator (which was introduced in \cite{MS22}) and
Leray-Lions method (c.f. \cite{BM92}, \cite{MT03}).
