We study a comparison principle and uniqueness of positive solutions for
the homogeneous Dirichlet boundary value problem associated to quasi-linear elliptic equations with
lower order terms. A model example is given by
$ -\Delta u+\lambda\frac{|\nabla u|^2}{u^r} = f(x), \qquad\lambda,r>0.$
The main feature of these equations consists in having a
quadratic gradient term in which singularities are allowed. The
arguments employed here also work to deal with equations having
lack of ellipticity or some dependence on u in the right hand
side.
Furthermore, they could be applied to obtain uniqueness results
for nonlinear equations having the p-Laplacian operator as the principal
part. Our results improve those already known, even if the gradient
term is not singular.