Uniqueness of solutions for some elliptic equations with a quadratic gradient term

Abstract

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 -Δ u+λ|∇u|2/ur = f(x), λ, 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. © EDP Sciences, SMAI, 2008.