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

¹ Departamento de Análisis Matemático, Universidad de Granada, Campus Fuentenueva s/n, 18071 Granada, Spain. darcoya@ugr.es
² Departament d'Anàlisi Matemàtica, Universitat de València, Dr. Moliner 50, 46100 Burjassot, Valencia, Spain. sergio.segura@uv.es

Received: 27 May 2008
Revised: 2 October 2008

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

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.