Some results on the qualitative properties of positive solutions of quasilinear elliptic equations

We consider the Dirichlet problem -Delta(m)- (u) = f (u) in Omega with zero Dirichlet boundary conditions. We prove local summability properties of 1/vertical bar Du vertical bar and we exploit these results to give geometric characterizations of the critical set Z = { x is an element of Omega vertical bar Du(x) = 0}. We extend to the case of changing sign nonlinearities some results known in the case f (s) > 0 for s > 0.