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Dirichlet problems with singular and gradient quadratic lower order terms

Published online by Cambridge University Press:  26 April 2008

Lucio Boccardo
Lucio Boccardo*
Affiliation:
Dipartimento di Matematica, Università di Roma 1, Piazza A. Moro 2, 00185 Roma, Italy; boccardo@mat.uniroma1.it
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Abstract

We present a revisited form of a resultproved in [Boccardo, Murat and Puel, Portugaliae Math.41 (1982) 507–534] and thenwe adapt the new proof in orderto show the existence for solutionsof quasilinear elliptic problems alsoif the lower order term has quadratic dependence on the gradient and singular dependence on the solution.

Keywords

Quadratic gradientsingular lower order term
Type
Research Article
Information
ESAIM: Control, Optimisation and Calculus of Variations , Volume 14 , Issue 3 , July 2008 , pp. 411 - 426
Copyright
© EDP Sciences, SMAI, 2008

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References

D. Arcoya, S. Barile and P.J. Martinez-Aparicio, Singular quasilinear equations with quadratic growth in the gradient without sign condition. Preprint.
D. Arcoya and P.J. Martinez-Aparicio, Quasilinear equations with natural growth Rev. Mat. Iberoamericana (to appear).
D. Arcoya, J. Carmona, T. Leonori, P.J. Martínez, L. Orsina and F. Petitta, Quadratic quasilinear equations with general singularities. Preprint.
Bensoussan, A., Boccardo, L. and Murat, F., On a nonlinear partial differential equation having natural growth terms and unbounded solution. Ann. Inst. H. Poincaré Anal. Non Linéaire 5 (1988) 347–364. CrossRef
L. Boccardo, Some nonlinear Dirichlet problems in $L\sp 1$ involving lower order terms in divergence form, in Progress in elliptic and parabolic partial differential equations (Capri, 1994), Pitman Res. Notes Math. Ser. 350, Longman, Harlow (1996) 43–57.
Boccardo, L., Positive solutions for some quasilinear elliptic equations with natural growths. Atti Accad. Naz. Lincei 11 (2000) 31–39.
L. Boccardo, Hardy potential and quasi-linear elliptic problems having natural growth terms, in Proceedings of the Conference held in Gaeta on the occasion of the 60th birthday of Haim Brezis, Progr. Nonlinear Differential Equations Appl. 63, Birkhauser, Basel (2005) 67–82.
Boccardo, L. and Gallouët, T., Nonlinear elliptic and parabolic equations involving measure data. J. Funct. Anal. 87 (1989) 149–169. CrossRef
Boccardo, L. and Gallouët, T., Strongly nonlinear elliptic equations having natural growth terms and L 1 data. Nonlinear Anal. 19 (1992) 573–579. CrossRef
Boccardo, L., Gallouët, T. and Orsina, L., Existence and nonexistence of solutions for some nonlinear elliptic equations. J. Anal. Math. 73 (1997) 203–223. CrossRef
Boccardo, L. and Giachetti, D., Existence results via regularity for some nonlinear elliptic problems. Comm. Partial Diff. Eq. 14 (1989) 663–680. CrossRef
Boccardo, L. and Murat, F., Almost everywhere convergence of the gradients of solutions to elliptic and parabolic equations. Nonlinear Anal. TMA 19 (1992) 581–597. CrossRef
Boccardo, L., Murat, F. and Puel, J.-P., Existence de solutions non bornées pour certaines équations quasi-linéaires. Portugaliae Math. 41 (1982) 507–534.
Boccardo, L., Murat, F. and Puel, J.-P., Résultats d'existence pour certains problèmes elliptiques quasi linéaires. Ann. Sc. Norm. Sup. Pisa 11 (1984) 213–235.
Boccardo, L., Murat, F. and Puel, J.-P., Existence of bounded solutions for nonlinear elliptic unilateral problems. Ann. Mat. Pura Appl. 152 (1988) 183–196. CrossRef
Boccardo, L., Murat, F. and Puel, J.-P., $L^\infty$ -estimates for some nonlinear partial differential equations and application to an existence result. SIAM J. Math. Anal. 23 (1992) 326–333. CrossRef
Brezis, H. and Nirenberg, L., Removable singularities for nonlinear elliptic equations. Topol. Methods Nonlinear Anal. 9 (1997) 201–219.
Crandall, M.G., Rabinowitz, P.H. and Tartar, L., On a Dirichlet problem with a singular nonlinearity. Comm. Partial Diff. Eq. 2 (1977) 193–222. CrossRef
Dall'Aglio, A., Giachetti, D. and Puel, J.-P., Nonlinear elliptic equations with natural growth in general domains. Ann. Mat. Pura Appl. 181 (2002) 407–426. CrossRef
Dall'Aglio, A., De Cicco, V., Giachetti, D. and Puel, J.-P., Existence of bounded solutions for nonlinear elliptic equations in unbounded domains. NoDEA 11 (2004) 431–450. CrossRef
Del Vecchio, T., Strongly nonlinear problems with Hamiltonian having natural growth. Houston J. Math. 16 (1990) 7–24.
D. Giachetti and F. Murat, Personal communication.
Keller, J.B., On solutions of $\Delta u=f(u)$ . Commun. Pure Appl. Math. 10 (1957) 503–510. CrossRef
Lazer, A.C. and McKenna, P.J., On a singular nonlinear elliptic boundary-value problem. Proc. Amer. Math. Soc. 111 (1991) 721–730. CrossRef
Leonori, T., Large solutions for a class of nonlinear elliptic equations with gradient terms. Adv. Nonlinear Stud. 7 (2007) 237–269.
Osserman, R., On the inequality $\Delta u\geq f(u)$ . Pacific J. Math. 7 (1957) 1641–1647. CrossRef
Porretta, A., Existence for elliptic equations in L 1 having lower order terms with natural growth. Portugaliae Math. 57 (2000) 179–190.
Porretta, A., A local estimates and large solutions for some elliptic equations with absorption. Adv. Differential Equations 9 (2004) 329–351.
Porretta, A. and Segura de, S. Leon, Nonlinear elliptic equations having a gradient term with natural growth. J. Math. Pures Appl. 85 (2006) 465–492. CrossRef
Puel, J.-P., Existence, comportement à l'infini et stabilité dans certains problèmes quasilinéaires elliptiques et paraboliques d'ordre 2. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 3 (1976) 89–119.
Stampacchia, G., Le problème de Dirichlet pour les équations elliptiques du second ordre à coefficients discontinus. Ann. Inst. Fourier (Grenoble) 15 (1965) 189–258. CrossRef
Trudinger, N.S., Harnack, On type inequalities and their application to quasilinear elliptic equations. Comm. Pure Appl. Math. 20 (1967) 721–747. CrossRef
J.L. Vazquez, The Porous Medium Equation: Mathematical Theory, Oxford Mathematical Monographs. Oxford University Press, Oxford (2007).