# 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

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topArcoya, 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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