UNIQUENESS OF SOLUTIONS FOR SOME ELLIPTIC EQUATIONS WITH A QUADRATIC GRADIENT TERM

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 vertical bar Delta u vertical bar(2)/u(r) = f(x), 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.