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}, zbl = {1189.35109}, mrnumber = {2654196}, language = {en}, url = {http://archive.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 DA - 2010/// SP - 327 EP - 336 VL - 16 IS - 2 PB - EDP-Sciences UR - http://archive.numdam.org/articles/10.1051/cocv:2008072/ UR - https://zbmath.org/?q=an%3A1189.35109 UR - https://www.ams.org/mathscinet-getitem?mr=2654196 UR - https://doi.org/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 https://doi.org/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://archive.numdam.org/articles/10.1051/cocv:2008072/

[1] Sobolev spaces. Academic Press, New York (1975). | Zbl

,[2] Existence results for nonlinear elliptic equations with degenerate coercivity. Ann. Mat. Pura Appl. (4) 182 (2003) 53-79. | Zbl

, , , and ,[3] Quasilinear equations with natural growth. Rev. Mat. Iberoamericana 24 (2008) 597-616. | Zbl

and ,[4] Elliptic obstacle problems with natural growth on the gradient and singular nonlinear terms. Adv. Nonlinear Stud. 7 (2007) 299-317. | Zbl

, and ,[5] Existence and nonwxistence of solutions for singular quadratic quasilinear equations. J. Differ. Equ. (submitted). | Zbl

, , , , and ,[6] Singular quasilinear equations with quadratic growth in the gradient without sign condition. J. Math. Anal. Appl. 350 (2009) 401-408. | Zbl

, and ,[7] Uniqueness and the maximum principle for quasilinear elliptic equations with quadratic growth conditions. Arch. Rational Mech. Anal. 133 (1995) 77-101. | Zbl

and ,[8] Uniqueness for unbounded solutions to stationary viscous Hamilton-Jacobi equations. Ann. Scuola Norm. Super. Pisa Cl. Sci. (5) 5 (2006) 107-136. | Numdam | Zbl

and ,[9] Remarks on the maximum principle for nonlinear elliptic PDEs with quadratic growth conditions. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 28 (1999) 381-404. | Numdam | Zbl

, , and ,[10] An L1-theory of existence and uniqueness of solutions of nonlinear elliptic equations. Ann. Scuola Norm Sup. Pisa Cl. Sci. (4) 22 (1995) 241-273. | Numdam | Zbl

, , , , , ,[11] Existence and uniqueness results for nonlinear elliptic problems with a lower order term and measure datum. C. R. Math. Acad. Sci. Paris 334 (2002) 757-762. | Zbl

, , and ,[12] Uniqueness of renormalized solutions to nonlinear elliptic equations with a lower order term and right hand side in L1(Ω). ESAIM: COCV 8 (2002) 239-272 | Numdam | Zbl

, , and ,[13] Uniqueness results for nonlinear elliptic equations with a lower order term. Nonlinear Anal. 63 (2005) 153-170. | Zbl

, , and ,[14] Quasi-linear degenerate elliptic problems with L1 data. Nonlinear Anal. 60 (2005) 557-587.

, and ,[15] Dirichlet problems with singular and gradient quadratic lower order terms. ESAIM: COCV 14 (2008) 411-426. | Numdam | Zbl

,[16] Existence and regularity of minima for integral functionals noncoercive in the energy space. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 25 (1997) 95-130. | Numdam | Zbl

and ,[17] Existence de solutions non bornées pour certaines équations quasi-linéaires. Portugal. Math. 41 (1982) 507-534. | Zbl

, and ,[18] Résultats d'existence pour certains problèmes elliptiques quasilinéaires. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 11 (1984) 213-235. | Numdam | Zbl

, and ,[19] Existence and regularity results for some elliptic equations with degenerate coercivity. Atti Sem. Mat. Fis. Univ. Modena 46 Suppl. (1998) 51-81. | Zbl

, and ,[20] Bounded and unbounded solutions for a class of quasi-linear elliptic problems with a quadratic gradient term. J. Math. Pures Appl. 80 (2001) 919-940. | Zbl

, and ,[21] Remarks on sublinear elliptic equations. Nonlinear Anal. T.M.A. 10 (1986) 55-64. | Zbl

and ,[22] Uniqueness of the Neumann condition and comparison results for Dirichlet pseudo-monotone problems, in The first 60 years of nonlinear analysis of Jean Mawhin, World Sci. Publ., River Edge, NJ (2004) 27-40. | Zbl

, and ,[23] Renormalized solutions of elliptic equations with general measure data. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 28 (1999) 741-808. | Numdam | Zbl

, , and ,[24] An elliptic problem with a lower order term having singular behaviour. Boll. Un. Mat. Ital. B (to appear). | Zbl

and ,[25] Elliptic Partial Differential Equations of Second Order. Springer-Verlag, New York (1983). | Zbl

and ,[26] Some qualitative properties of solutions of quasilinear elliptic equations and applications. J. Differ. Equ. 170 (2001) 247-280. | Zbl

, and ,[27] Uniqueness of solutions of some elliptic equations without condition at infinity. C. R. Math. Acad. Sci. Paris 335 (2002) 739-744. | Zbl

,[28] Some uniqueness results for elliptic equations without condition at infinity. Commun. Contemp. Math. 5 (2003) 705-717. | Zbl

,[29] Uniqueness of solutions for some nonlinear Dirichlet problems. NoDEA Nonlinear Differ. Equ. Appl. 11 (2004) 407-430. | Zbl

,[30] Nonlinear elliptic equations having a gradient term with natural growth. J. Math. Pures Appl. 85 (2006) 465-492. | Zbl

and ,[31] Existence and uniqueness for L1 data of some elliptic equations with natural growth. Adv. Differential Equations 8 (2003) 1377-1408. | Zbl

,*Cited by Sources: *