  In this paper, we prove existence and regularity results for solutions of
some nonlinear Dirichlet problems for an elliptic equation defined by a
degenerate coercive operator and a singular right hand side.
\begin{equation}\label{01} \left\{ \begin{array}{lll} -\displaystyle\mbox{div}(
a(x,u,\nabla u))&=\displaystyle\frac{f}{u^{\gamma}} & \mbox{ in } \Omega \\
u&>0 &\mbox{ in }\Omega \\ u&=0 &\mbox{ on } \delta\Omega \end{array} \right.
\end{equation} where $\Omega $ is bounded open subset of $I\!\!R^{N}(N\geq2),$
$\gamma>0$ and $ f$ is a nonnegative function that belongs to some Lebesgue
space.
