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

David Arcoya; Sergio Segura de León

ESAIM: Control, Optimisation and Calculus of Variations (2010)

  • Volume: 16, Issue: 2, page 327-336
  • ISSN: 1292-8119

Abstract

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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 u r = 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.

How to cite

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Arcoya, David, and Segura de León, Sergio. "Uniqueness of solutions for some elliptic equations with a quadratic gradient term." ESAIM: Control, Optimisation and Calculus of Variations 16.2 (2010): 327-336. <http://eudml.org/doc/250798>.

@article{Arcoya2010,
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.},
author = {Arcoya, David, Segura de León, Sergio},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
keywords = {Non linear elliptic problems; uniqueness; comparison principle; lower order terms with singularities at the Gradient term; lack of coerciveness; non linear elliptic problems; lower order terms with singularities at the gradient term},
language = {eng},
month = {4},
number = {2},
pages = {327-336},
publisher = {EDP Sciences},
title = {Uniqueness of solutions for some elliptic equations with a quadratic gradient term},
url = {http://eudml.org/doc/250798},
volume = {16},
year = {2010},
}

TY - JOUR
AU - Arcoya, David
AU - Segura de León, Sergio
TI - Uniqueness of solutions for some elliptic equations with a quadratic gradient term
JO - ESAIM: Control, Optimisation and Calculus of Variations
DA - 2010/4//
PB - EDP Sciences
VL - 16
IS - 2
SP - 327
EP - 336
AB - 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.
LA - eng
KW - Non linear elliptic problems; uniqueness; comparison principle; lower order terms with singularities at the Gradient term; lack of coerciveness; non linear elliptic problems; lower order terms with singularities at the gradient term
UR - http://eudml.org/doc/250798
ER -

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