Degenerate elliptic problem with a singular nonlinearity

Abstract: 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.