  We consider a boundary value problem in a bounded domain involving a
degenerate operator of the form $$L(u)=-\textrm{div} (a(x)\nabla u)$$ and a
suitable nonlinearity $f$. The function $a$ vanishes on smooth 1-codimensional
submanifolds of $\Omega$ where it is not allowed to be $C^{2}$. By using
weighted Sobolev spaces we are still able to find existence of solutions which
vanish, in the trace sense, on the set where $a$ vanishes.
