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

Arcoya, David  ; Segura de León, Sergio

ESAIM: Control, Optimisation and Calculus of Variations, Tome 16 (2010) no. 2, pp. 327-336

Résumé

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 $-\Delta u+\lambda \frac{{\left|\nabla u\right|}^2}{u^r}=f\left(x\right),\lambda ,r>0.$ 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.

Détail

DOI : 10.1051/cocv:2008072

Classification : 35J65, 35J70, 35J60
Keywords: non linear elliptic problems, uniqueness, comparison principle, lower order terms with singularities at the gradient term, lack of coerciveness