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

Published online by Cambridge University Press:  19 December 2008

David Arcoya  and
Sergio Segura de León
David Arcoya
Affiliation:
Departamento de Análisis Matemático, Universidad de Granada, Campus Fuentenueva s/n, 18071 Granada, Spain. darcoya@ugr.es
Sergio Segura de León
Affiliation:
Departament d'Anàlisi Matemàtica, Universitat de València, Dr. Moliner 50, 46100 Burjassot, Valencia, Spain. sergio.segura@uv.es
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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

$ -\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.

Keywords

Non linear elliptic problemsuniquenesscomparison principlelower order terms with singularities at the Gradient termlack of coerciveness
Type
Research Article
Information
ESAIM: Control, Optimisation and Calculus of Variations , Volume 16 , Issue 2 , April 2010 , pp. 327 - 336
Copyright
© EDP Sciences, SMAI, 2008

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