Comptes Rendus
Partial Differential Equations
Correlation between two quasilinear elliptic problems with a source term involving the function or its gradient
[Corrélation entre deux problèmes quasilinéaires elliptiques avec terme de source relatif à la fonction ou à son gradient]
Comptes Rendus. Mathématique, Volume 346 (2008) no. 23-24, pp. 1251-1256.

Thanks to a change of unknown we compare two elliptic quasilinear problems with Dirichlet data in a bounded domain of RN. The first one, of the form Δpu=β(u)|u|p+λf(x), where β is nonnegative, involves a gradient term with natural growth. The second one, of the form Δpv=λf(x)(1+g(v))p1 where g is nondecreasing, presents a source term of order 0. The correlation gives new results of existence, nonexistence and multiplicity for the two problems.

A l'aide d'un changement d'inconnue nous comparons deux problèmes elliptiques quasilinéaires avec conditions de Dirichlet dans un domaine borné Ω de RN. Le premier, de la forme Δpu=β(u)|u|p+λf(x), où β est positif, comporte un terme de gradient à croissance critique. Le second, de la forme Δpv=λf(x)(1+g(v))p1g est croissante, contient un terme de source d'ordre 0. La comparaison donne des résultats nouveaux d'existence, nonexistence et multiplicité pour les deux problèmes.

Reçu le :
Publié le :
DOI : 10.1016/j.crma.2008.10.002

Haydar Abdel Hamid 1 ; Marie Françoise Bidaut-Véron 1

1 Laboratoire de mathématiques et physique théorique, CNRS UMR 6083, faculté des sciences, 37200 Tours, France
@article{CRMATH_2008__346_23-24_1251_0,
     author = {Haydar Abdel Hamid and Marie Fran\c{c}oise Bidaut-V\'eron},
     title = {Correlation between two quasilinear elliptic problems with a source term involving the function or its gradient},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {1251--1256},
     publisher = {Elsevier},
     volume = {346},
     number = {23-24},
     year = {2008},
     doi = {10.1016/j.crma.2008.10.002},
     language = {en},
}
TY  - JOUR
AU  - Haydar Abdel Hamid
AU  - Marie Françoise Bidaut-Véron
TI  - Correlation between two quasilinear elliptic problems with a source term involving the function or its gradient
JO  - Comptes Rendus. Mathématique
PY  - 2008
SP  - 1251
EP  - 1256
VL  - 346
IS  - 23-24
PB  - Elsevier
DO  - 10.1016/j.crma.2008.10.002
LA  - en
ID  - CRMATH_2008__346_23-24_1251_0
ER  - 
%0 Journal Article
%A Haydar Abdel Hamid
%A Marie Françoise Bidaut-Véron
%T Correlation between two quasilinear elliptic problems with a source term involving the function or its gradient
%J Comptes Rendus. Mathématique
%D 2008
%P 1251-1256
%V 346
%N 23-24
%I Elsevier
%R 10.1016/j.crma.2008.10.002
%G en
%F CRMATH_2008__346_23-24_1251_0
Haydar Abdel Hamid; Marie Françoise Bidaut-Véron. Correlation between two quasilinear elliptic problems with a source term involving the function or its gradient. Comptes Rendus. Mathématique, Volume 346 (2008) no. 23-24, pp. 1251-1256. doi : 10.1016/j.crma.2008.10.002. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2008.10.002/

[1] B. Abdellaoui; A. Dall'Aglio; I. Peral Some remarks on elliptic problems with critical growth in the gradient, J. Differential Equations, Volume 222 (2006), pp. 21-62

[2] H. Brezis; T. Cazenave; Y. Martel; A. Ramiandrisoa Blow-up for utΔu=g(u) revisited, Adv. Differential Equations, Volume 1 (1996), pp. 73-90

[3] X. Cabre; M. Sanchon Semi-stable and extremal solutions of reaction equations involving the p-Laplacian, Comm. Pure Appl. Anal., Volume 6 (2007), pp. 43-67

[4] G. Dal Maso; F. Murat; L. Orsina; A. Prignet Renormalized solutions of elliptic equations with general measure data, Ann. Scuola Norm. Sup. Pisa, Volume 28 (1999), pp. 741-808

[5] A. Ferrero On the solutions of quasilinear elliptic equations with a polynomial-type reaction term, Adv. Differential Equations, Volume 9 (2004), pp. 1201-1234

[6] V. Ferone; F. Murat Nonlinear problems having natural growth in the gradient: an existence result when the source terms are small, Nonlinear Anal., Volume 42 (2000), pp. 1309-1326

[7] N. Ghoussoub; D. Preiss A general mountain path principle for locating and classifying critical points, Ann. Inst. H. Poincaré Anal. Non Linéaire, Volume 6 (1989) no. 5, pp. 321-330

[8] L. Jeanjean On the existence of bounded Palais–Smale sequences and application to a Landesman–Lazer type problem, Proc. Roy. Soc. Edinburgh Sect. A, Volume 129 (1999), pp. 787-809

[9] G. Nedev Regularity of the extremal solution of semilinear elliptic equations, C. R. Acad. Sci. Paris Sér. I Math., Volume 330 (2000), pp. 997-2002

[10] A. Porretta Nonlinear equations with natural growth terms and measure data, Electron. J. Differential Equations Conf., Volume 9 (2002), pp. 183-202

[11] M. Sanchon Boundedness of the extremal solutions of some p-Laplacian problems, Nonlinear Anal., Volume 67 (2007), pp. 281-294

  • J.V. Goncalves; M.R. Marcial; O.H. Miyagaki; C.A.P. dos Santos Topological structure of the set of solution for singular elliptic equations with a convective term, Journal of Mathematical Analysis and Applications, Volume 552 (2025) no. 2, p. 129738 | DOI:10.1016/j.jmaa.2025.129738
  • Boumediene Abdellaoui; Andrea Dall’Aglio; Sergio Segura de León Multiplicity of Solutions to Elliptic Problems Involving the 1-Laplacian with a Critical Gradient Term, Advanced Nonlinear Studies, Volume 17 (2017) no. 2, p. 333 | DOI:10.1515/ans-2017-0011
  • François Hamel; Emmanuel Russ Comparison results and improved quantified inequalities for semilinear elliptic equations, Mathematische Annalen, Volume 367 (2017) no. 1-2, p. 311 | DOI:10.1007/s00208-016-1394-1
  • Luiz F. O. Faria; Olímpio H. Miyagaki; Fábio R. Pereira Quasilinear elliptic system in exterior domains with dependence on the gradient, Mathematische Nachrichten, Volume 287 (2014) no. 4, p. 361 | DOI:10.1002/mana.201100006
  • Louis Jeanjean; Boyan Sirakov Existence and Multiplicity for Elliptic Problems with Quadratic Growth in the Gradient, Communications in Partial Differential Equations, Volume 38 (2013) no. 2, p. 244 | DOI:10.1080/03605302.2012.738754
  • Marco Magliaro; Luciano Mari; Paolo Mastrolia; Marco Rigoli Keller–Osserman type conditions for differential inequalities with gradient terms on the Heisenberg group, Journal of Differential Equations, Volume 250 (2011) no. 6, p. 2643 | DOI:10.1016/j.jde.2011.01.006
  • ROBERTA FILIPPUCCI; PATRIZIA PUCCI; MARCO RIGOLI NONLINEAR WEIGHTED p-LAPLACIAN ELLIPTIC INEQUALITIES WITH GRADIENT TERMS, Communications in Contemporary Mathematics, Volume 12 (2010) no. 03, p. 501 | DOI:10.1142/s0219199710003841

Cité par 7 documents. Sources : Crossref

Commentaires - Politique