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+β(x)Ο{s>0}β+Aββ(x)Ο{sβ€0}β and f(x,s)=f+β(x)Ο{s>0}β+fββ(x)Ο{sβ€0}β. Under the ellipticity assumption
that ΞΌ1ββ€AΒ±ββ€ΞΌ, 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 W1,p(B1β) which is also Cloc0,Ξ±β for every
Ξ±β(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})