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 $-\Delta u+\lambda \frac{{\left|\nabla u\right|}^{2}}{{u}^{r}}=f\left(x\right),\phantom{\rule{2em}{0ex}}\lambda ,r>0.$ 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 problems, uniqueness, comparison principle, lower order terms with singularities at the gradient term, lack of coerciveness

@article{COCV_2010__16_2_327_0, author = {Arcoya, David and Segura de Le\'on, Sergio}, title = {Uniqueness of solutions for some elliptic equations with a quadratic gradient term}, journal = {ESAIM: Control, Optimisation and Calculus of Variations}, pages = {327--336}, publisher = {EDP-Sciences}, volume = {16}, number = {2}, year = {2010}, doi = {10.1051/cocv:2008072}, mrnumber = {2654196}, zbl = {1189.35109}, language = {en}, url = {http://www.numdam.org/articles/10.1051/cocv:2008072/} }

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 PY - 2010 SP - 327 EP - 336 VL - 16 IS - 2 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/cocv:2008072/ DO - 10.1051/cocv:2008072 LA - en ID - COCV_2010__16_2_327_0 ER -

%0 Journal Article %A Arcoya, David %A Segura de León, Sergio %T Uniqueness of solutions for some elliptic equations with a quadratic gradient term %J ESAIM: Control, Optimisation and Calculus of Variations %D 2010 %P 327-336 %V 16 %N 2 %I EDP-Sciences %U http://www.numdam.org/articles/10.1051/cocv:2008072/ %R 10.1051/cocv:2008072 %G en %F COCV_2010__16_2_327_0

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://www.numdam.org/articles/10.1051/cocv:2008072/

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