We consider a nonlinear Dirichlet problem driven by thep-Laplacian, a convection term and a(p - 1)-sublinear perturbation. First we assume that the coefficient in the convection term (drift coefficient) is sign changing. Using the theory of nonlinear operators of monotone type together with suitable truncation and comparison techniques we prove the existence of a positive smooth solution. When the drift coefficient is nonnegative, we are able to relax the conditions on the data of the problem.