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 $-\Delta u+\lambda \frac{{|\nabla u|}^{2}}{{u}^{r}}=f\left(x\right),\phantom{\rule{2em}{0ex}}\lambda ,r>0.$ 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.

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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author = {Arcoya, David and Segura de Le\'on, Sergio},
title = {Uniqueness of solutions for some elliptic equations with a quadratic gradient term},
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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/

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

[2] A. Alvino, L. Boccardo, V. Ferone, L. Orsina and G. Trombetti, Existence results for nonlinear elliptic equations with degenerate coercivity. Ann. Mat. Pura Appl. (4) 182 (2003) 53-79. | Zbl

[3] D. Arcoya and P.J. Martínez-Aparicio, Quasilinear equations with natural growth. Rev. Mat. Iberoamericana 24 (2008) 597-616. | Zbl

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

[5] D. Arcoya, J. Carmona, T. Leonori, P.J. Martínez-Aparicio, L. Orsina and F. Petitta, Existence and nonwxistence of solutions for singular quadratic quasilinear equations. J. Differ. Equ. (submitted). | Zbl

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

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

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

[9] G. Barles, A.P. Blanc, C. Georgelin and M. Kobylanski, 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

[10] P. Bénilan, L. Boccardo, T. Gallouët, R. Gariepy, M. Pierre, J.L. Vázquez, 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] M.F. Betta, A. Mercaldo, F. Murat and M.M. Porzio, 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

[12] M.F. Betta, A. Mercaldo, F. Murat and M.M. Porzio, 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

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

[14] D. Blanchard, F. Désir and O. Guibé, Quasi-linear degenerate elliptic problems with L1 data. Nonlinear Anal. 60 (2005) 557-587.

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

[16] L. Boccardo and L. Orsina, 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

[17] L. Boccardo, F. Murat and J.P. Puel, Existence de solutions non bornées pour certaines équations quasi-linéaires. Portugal. Math. 41 (1982) 507-534. | Zbl

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

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

[20] L. Boccardo, S. Segura De León and C. Trombetti, 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

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

[22] J. Casado-Díaz, F. Murat and A. Porretta, 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

[23] G. Dal Maso, F. Murat, L. Orsina and A. Prignet, Renormalized solutions of elliptic equations with general measure data. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 28 (1999) 741-808. | Numdam | Zbl

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

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

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

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

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

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

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

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

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