Dirichlet problems with singular and gradient quadratic lower order terms

Lucio Boccardo

ESAIM: Control, Optimisation and Calculus of Variations (2008)

  • Volume: 14, Issue: 3, page 411-426
  • ISSN: 1292-8119

Abstract

top
We present a revisited form of a result proved in [Boccardo, Murat and Puel, Portugaliae Math.41 (1982) 507–534] and then we adapt the new proof in order to show the existence for solutions of quasilinear elliptic problems also if the lower order term has quadratic dependence on the gradient and singular dependence on the solution.

How to cite

top

Boccardo, Lucio. "Dirichlet problems with singular and gradient quadratic lower order terms." ESAIM: Control, Optimisation and Calculus of Variations 14.3 (2008): 411-426. <http://eudml.org/doc/250307>.

@article{Boccardo2008,
abstract = { We present a revisited form of a result proved in [Boccardo, Murat and Puel, Portugaliae Math.41 (1982) 507–534] and then we adapt the new proof in order to show the existence for solutions of quasilinear elliptic problems also if the lower order term has quadratic dependence on the gradient and singular dependence on the solution. },
author = {Boccardo, Lucio},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
keywords = {Quadratic gradient; singular lower order term; quadratic gradient; Dirichlet problems; quasilinear elliptic problems},
language = {eng},
month = {4},
number = {3},
pages = {411-426},
publisher = {EDP Sciences},
title = {Dirichlet problems with singular and gradient quadratic lower order terms},
url = {http://eudml.org/doc/250307},
volume = {14},
year = {2008},
}

TY - JOUR
AU - Boccardo, Lucio
TI - Dirichlet problems with singular and gradient quadratic lower order terms
JO - ESAIM: Control, Optimisation and Calculus of Variations
DA - 2008/4//
PB - EDP Sciences
VL - 14
IS - 3
SP - 411
EP - 426
AB - We present a revisited form of a result proved in [Boccardo, Murat and Puel, Portugaliae Math.41 (1982) 507–534] and then we adapt the new proof in order to show the existence for solutions of quasilinear elliptic problems also if the lower order term has quadratic dependence on the gradient and singular dependence on the solution.
LA - eng
KW - Quadratic gradient; singular lower order term; quadratic gradient; Dirichlet problems; quasilinear elliptic problems
UR - http://eudml.org/doc/250307
ER -

References

top
  1. D. Arcoya, S. Barile and P.J. Martinez-Aparicio, Singular quasilinear equations with quadratic growth in the gradient without sign condition. Preprint.  Zbl1161.35013
  2. D. Arcoya and P.J. Martinez-Aparicio, Quasilinear equations with natural growth Rev. Mat. Iberoamericana (to appear). Zbl1151.35343
  3. D. Arcoya, J. Carmona, T. Leonori, P.J. Martínez, L. Orsina and F. Petitta, Quadratic quasilinear equations with general singularities. Preprint.  Zbl1173.35051
  4. A. Bensoussan, L. Boccardo and F. Murat, On a nonlinear partial differential equation having natural growth terms and unbounded solution. Ann. Inst. H. Poincaré Anal. Non Linéaire5 (1988) 347–364.  Zbl0696.35042
  5. L. Boccardo, Some nonlinear Dirichlet problems in L 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.  Zbl0889.35034
  6. L. Boccardo, Positive solutions for some quasilinear elliptic equations with natural growths. Atti Accad. Naz. Lincei11 (2000) 31–39.  Zbl0970.35061
  7. 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.  Zbl1124.35019
  8. L. Boccardo and T. Gallouët, Nonlinear elliptic and parabolic equations involving measure data. J. Funct. Anal.87 (1989) 149–169.  Zbl0707.35060
  9. L. Boccardo and T. Gallouët, Strongly nonlinear elliptic equations having natural growth terms and L1 data. Nonlinear Anal.19 (1992) 573–579.  Zbl0795.35031
  10. L. Boccardo, T. Gallouët and L. Orsina, Existence and nonexistence of solutions for some nonlinear elliptic equations. J. Anal. Math.73 (1997) 203–223.  Zbl0898.35035
  11. L. Boccardo and D. Giachetti, Existence results via regularity for some nonlinear elliptic problems. Comm. Partial Diff. Eq.14 (1989) 663–680.  Zbl0678.35035
  12. L. Boccardo and F. Murat, Almost everywhere convergence of the gradients of solutions to elliptic and parabolic equations. Nonlinear Anal. TMA19 (1992) 581–597.  Zbl0783.35020
  13. L. Boccardo, F. Murat and J.-P. Puel, Existence de solutions non bornées pour certaines équations quasi-linéaires. Portugaliae Math.41 (1982) 507–534.  Zbl0524.35041
  14. L. Boccardo, F. Murat and J.-P. Puel, Résultats d'existence pour certains problèmes elliptiques quasi linéaires. Ann. Sc. Norm. Sup. Pisa11 (1984) 213–235.  Zbl0557.35051
  15. L. Boccardo, F. Murat and J.-P. Puel, Existence of bounded solutions for nonlinear elliptic unilateral problems. Ann. Mat. Pura Appl.152 (1988) 183–196.  Zbl0687.35042
  16. L. Boccardo, F. Murat and J.-P. Puel, L -estimates for some nonlinear partial differential equations and application to an existence result. SIAM J. Math. Anal.23 (1992) 326–333.  Zbl0785.35033
  17. H. Brezis and L. Nirenberg, Removable singularities for nonlinear elliptic equations. Topol. Methods Nonlinear Anal.9 (1997) 201–219.  Zbl0905.35027
  18. M.G. Crandall, P.H. Rabinowitz and L. Tartar, On a Dirichlet problem with a singular nonlinearity. Comm. Partial Diff. Eq.2 (1977) 193–222.  Zbl0362.35031
  19. A. Dall'Aglio, D. Giachetti and J.-P. Puel, Nonlinear elliptic equations with natural growth in general domains. Ann. Mat. Pura Appl.181 (2002) 407–426.  Zbl1097.35050
  20. A. Dall'Aglio, V. De Cicco, D. Giachetti and J.-P. Puel, Existence of bounded solutions for nonlinear elliptic equations in unbounded domains. NoDEA11 (2004) 431–450.  Zbl1120.35038
  21. T. Del Vecchio, Strongly nonlinear problems with Hamiltonian having natural growth. Houston J. Math.16 (1990) 7–24.  Zbl0714.35035
  22. D. Giachetti and F. Murat, Personal communication.  
  23. J.B. Keller, On solutions of Δ u = f ( u ) . Commun. Pure Appl. Math.10 (1957) 503–510.  Zbl0090.31801
  24. A.C. Lazer and P.J. McKenna, On a singular nonlinear elliptic boundary-value problem. Proc. Amer. Math. Soc.111 (1991) 721–730.  Zbl0727.35057
  25. T. Leonori, Large solutions for a class of nonlinear elliptic equations with gradient terms. Adv. Nonlinear Stud.7 (2007) 237–269.  Zbl1156.35030
  26. R. Osserman, On the inequality Δ u f ( u ) . Pacific J. Math.7 (1957) 1641–1647.  Zbl0083.09402
  27. A. Porretta, Existence for elliptic equations in L1 having lower order terms with natural growth. Portugaliae Math.57 (2000) 179–190.  Zbl0963.35068
  28. A. Porretta, A local estimates and large solutions for some elliptic equations with absorption. Adv. Differential Equations9 (2004) 329–351.  Zbl1150.35401
  29. A. Porretta and S. Segura de Leon, Nonlinear elliptic equations having a gradient term with natural growth. J. Math. Pures Appl.85 (2006) 465–492.  Zbl1158.35364
  30. J.-P. Puel, 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.  Zbl0331.35027
  31. G. Stampacchia, Le problème de Dirichlet pour les équations elliptiques du second ordre à coefficients discontinus. Ann. Inst. Fourier (Grenoble)15 (1965) 189–258.  Zbl0151.15401
  32. N.S. Trudinger, On Harnack type inequalities and their application to quasilinear elliptic equations. Comm. Pure Appl. Math.20 (1967) 721–747.  Zbl0153.42703
  33. J.L. Vazquez, The Porous Medium Equation: Mathematical Theory, Oxford Mathematical Monographs. Oxford University Press, Oxford (2007).  

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.