Uniqueness of solutions for some elliptic equations with a quadratic gradient term
ESAIM: Control, Optimisation and Calculus of Variations, Volume 16 (2010) no. 2, pp. 327-336.

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 -Δu+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.

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
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     title = {Uniqueness of solutions for some elliptic equations with a quadratic gradient term},
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     pages = {327--336},
     publisher = {EDP-Sciences},
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Arcoya, David; Segura de León, Sergio. Uniqueness of solutions for some elliptic equations with a quadratic gradient term. ESAIM: Control, Optimisation and Calculus of Variations, Volume 16 (2010) no. 2, pp. 327-336. doi : 10.1051/cocv:2008072. http://archive.numdam.org/articles/10.1051/cocv:2008072/

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