Anisotropic problem with singular nonlinearity

Using an approximation approach, we prove the existence and regularity of the solutions to the following anisotropic problem, involving a singular nonlinearity N {-Sigma partial derivative(i) [vertical bar partial derivative(i)u vertical bar(pi-2) partial derivative(i)u] = f/u(gamma) in Omega, i = 1 u = 0 on Omega, u >= 0 in Omega, where Omega is a bounded regular domain in IRN and gamma > 0, we will assume without loss of generality that 1 <= p1 <= p2 <= . . . <= pN and that f is a non-negative function belonging to a suitable Lebesgue space L-m (Omega).