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Uniqueness of solutions for some elliptic equations with a quadratic gradient term

Published online by Cambridge University Press:  19 December 2008

David Arcoya  and
Sergio Segura de León
David Arcoya
Affiliation:
Departamento de Análisis Matemático, Universidad de Granada, Campus Fuentenueva s/n, 18071 Granada, Spain. [email protected]
Sergio Segura de León
Affiliation:
Departament d'Anàlisi Matemàtica, Universitat de València, Dr. Moliner 50, 46100 Burjassot, Valencia, Spain. [email protected]
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Abstract

We study a comparison principle and uniqueness of positive solutions forthe homogeneous Dirichlet boundary value problem associated to quasi-linear elliptic equations withlower 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 handside. Furthermore, they could be applied to obtain uniqueness results for nonlinear equations having the p-Laplacian operator as the principalpart. Our results improve those already known, even if the gradient term is not singular.

Keywords

Non linear elliptic problemsuniquenesscomparison principlelower order terms with singularities at the Gradient termlack of coerciveness
Type
Research Article
Information
ESAIM: Control, Optimisation and Calculus of Variations , Volume 16 , Issue 2 , April 2010 , pp. 327 - 336
Copyright
© EDP Sciences, SMAI, 2008

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References

R.A. Adams, Sobolev spaces. Academic Press, New York (1975).
Alvino, A., Boccardo, L., Ferone, V., Orsina, L. and Trombetti, G., Existence results for nonlinear elliptic equations with degenerate coercivity. Ann. Mat. Pura Appl. (4) 182 (2003) 53–79. CrossRef
Arcoya, D. and Martínez-Aparicio, P.J., Quasilinear equations with natural growth. Rev. Mat. Iberoamericana 24 (2008) 597–616. CrossRef
Arcoya, D., Carmona, J. and Martínez-Aparicio, P.J., Elliptic obstacle problems with natural growth on the gradient and singular nonlinear terms. Adv. Nonlinear Stud. 7 (2007) 299–317. CrossRef
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).
Arcoya, D., Barile, S. and Martínez-Aparicio, P.J., Singular quasilinear equations with quadratic growth in the gradient without sign condition. J. Math. Anal. Appl. 350 (2009) 401–408. CrossRef
Barles, G. and Murat, F., Uniqueness and the maximum principle for quasilinear elliptic equations with quadratic growth conditions. Arch. Rational Mech. Anal. 133 (1995) 77–101. CrossRef
Barles, G. and Porretta, A., Uniqueness for unbounded solutions to stationary viscous Hamilton-Jacobi equations. Ann. Scuola Norm. Super. Pisa Cl. Sci. (5) 5 (2006) 107–136.
Barles, G., Blanc, A.P., Georgelin, C. and Kobylanski, M., 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.
Bénilan, P., Boccardo, L., Gallouët, T., Gariepy, R., Pierre, M., Vázquez, J.L., 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.
Betta, M.F., Mercaldo, A., Murat, F. and Porzio, M.M., 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. CrossRef
Betta, M.F., Mercaldo, A., Murat, F. and Porzio, M.M., Uniqueness of renormalized solutions to nonlinear elliptic equations with a lower order term and right hand side in L 1(Ω). ESAIM: COCV 8 (2002) 239–272 CrossRef
Betta, M.F., Mercaldo, A., Murat, F. and Porzio, M.M., Uniqueness results for nonlinear elliptic equations with a lower order term. Nonlinear Anal. 63 (2005) 153–170. CrossRef
Blanchard, D., Désir, F. and Guibé, O., Quasi-linear degenerate elliptic problems with L1 data. Nonlinear Anal. 60 (2005) 557–587.
Boccardo, L., Dirichlet problems with singular and gradient quadratic lower order terms. ESAIM: COCV 14 (2008) 411–426. CrossRef
Boccardo, L. and Orsina, L., 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.
Boccardo, L., Murat, F. and Puel, J.P., Existence de solutions non bornées pour certaines équations quasi-linéaires. Portugal. Math. 41 (1982) 507–534.
Boccardo, L., Murat, F. and Puel, J.P., Résultats d'existence pour certains problèmes elliptiques quasilinéaires. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 11 (1984) 213–235.
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.
Boccardo, L., Segura de, S. 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. CrossRef
Brezis, H. and Oswald, L., Remarks on sublinear elliptic equations. Nonlinear Anal. T.M.A. 10 (1986) 55–64. CrossRef
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.
Dal Maso, G., Murat, F., Orsina, L. and Prignet, A., Renormalized solutions of elliptic equations with general measure data. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 28 (1999) 741–808.
D. Giachetti and F. Murat, An elliptic problem with a lower order term having singular behaviour. Boll. Un. Mat. Ital. B (to appear).
D. Gilbarg and N.S. Trudinger, Elliptic Partial Differential Equations of Second Order. Springer-Verlag, New York (1983).
Korkut, L., Pašić, M. and D. Žubrinić, Some qualitative properties of solutions of quasilinear elliptic equations and applications. J. Differ. Equ. 170 (2001) 247–280. CrossRef
Porretta, A., Uniqueness of solutions of some elliptic equations without condition at infinity. C. R. Math. Acad. Sci. Paris 335 (2002) 739–744. CrossRef
Porretta, A., Some uniqueness results for elliptic equations without condition at infinity. Commun. Contemp. Math. 5 (2003) 705–717. CrossRef
Porretta, A., Uniqueness of solutions for some nonlinear Dirichlet problems. NoDEA Nonlinear Differ. Equ. Appl. 11 (2004) 407–430. CrossRef
Porretta, A. and Segura de, S. León, Nonlinear elliptic equations having a gradient term with natural growth. J. Math. Pures Appl. 85 (2006) 465–492. CrossRef
Segura de, S. León, Existence and uniqueness for L 1 data of some elliptic equations with natural growth. Adv. Differential Equations 8 (2003) 1377–1408.