Singular elliptic problem with super-quadratic growth in the gradient

In this paper we analyze the existence of solution to the homogeneous Dirichlet for a nonlinear elliptic singular problem, {-Delta u = vertical bar del u vertical bar(q)/vertical bar u vertical bar(alpha) + lambda f in Omega, u = 0 on partial derivative Omega, where N >= 2, Omega subset of R-N is a bounded regular domain, q > 2, alpha < q/2, lambda > 0 and f is a nonnegative function belonging to a suitable Lebesgue space. Under additional hypotheses on the data, we are also able to show the uniqueness of the 'good' solution.