Localization of solutions for nonlinear elliptic problems with critical growth

Rejeb Hadiji; Riccardo Molle; Donato Passaseo; Habib Yazidi

doi:10.1016/j.crma.2006.10.018

Partial Differential Equations

Localization of solutions for nonlinear elliptic problems with critical growth
[Localisation des solutions pour un problème elliptique avec exposant critique de Sobolev]

Présenté par : Haïm Brezis

Rejeb Hadiji ¹ ; Riccardo Molle ² ; Donato Passaseo ³ ; Habib Yazidi ¹

¹ UFR des sciences et technologie, CNRS UMR 8050, université Paris 12, Val-de-Marne, 61, avenue du Général de Gaulle, 94010 Créteil, France
² Dipartimento di Matematica Università di Roma “Tor Vergata”, Via della Ricerca Scientifica, 1, 00133 Roma, Italy
³ Dipartimento di Matematica “E. De Giorgi”, Università di Lecce, P.O. Box 193, 73100 Lecce, Italy

Comptes Rendus. Mathématique, Volume 343 (2006) no. 11-12, pp. 725-730.

Résumé
Abstract

On étudie l'existence et la multiplicité de solutions du problème $- div (p (x) \nabla u) = u^{2^{*} - 1} + λ u$ , $u > 0$ dans Ω et $u = 0$ sur ∂Ω dans le cas où l'ensemble de minima de p admet plusieurs composantes connexes. On s'intéresse également au cas où cet ensemble possède une seule composante connexe et une topologie complexe.

We study the existence and the multiplicity of solutions for the problem $- div (p (x) \nabla u) = u^{2^{*} - 1} + λ u$ , $u > 0$ in Ω and $u = 0$ on ∂Ω, when the set of the minimizers for the weight p has multiple connected component. We study also the case where this set has one connected component and has complex topology.

Accepté le : 2006-10-15
Publié le : 2006-11-20

DOI : 10.1016/j.crma.2006.10.018

Affiliations des auteurs :

Rejeb Hadiji ¹ ; Riccardo Molle ² ; Donato Passaseo ³ ; Habib Yazidi ¹

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     author = {Rejeb Hadiji and Riccardo Molle and Donato Passaseo and Habib Yazidi},
     title = {Localization of solutions for nonlinear elliptic problems with critical growth},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {725--730},
     publisher = {Elsevier},
     volume = {343},
     number = {11-12},
     year = {2006},
     doi = {10.1016/j.crma.2006.10.018},
     language = {en},
}

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Rejeb Hadiji; Riccardo Molle; Donato Passaseo; Habib Yazidi. Localization of solutions for nonlinear elliptic problems with critical growth. Comptes Rendus. Mathématique, Volume 343 (2006) no. 11-12, pp. 725-730. doi : 10.1016/j.crma.2006.10.018. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2006.10.018/

Bibliographie
Cité par

[1] S. Alama; G. Tarantello Elliptic problems with nonlinearities indefinite in sign, J. Funct. Anal., Volume 141 (1996), pp. 159-214

[2] A. Ambrosetti; P. Rabinowitz Dual variational methods in critical point theory and applications, J. Funct. Anal., Volume 14 (1973), pp. 349-381

[3] H. Brezis Elliptic equations with limiting Sobolev exponents – the impact of topology, Comm. Pure Appl. Math., Volume 39 (1986), p. S17-S39

[4] H. Brezis; E. Lieb A relation between pointwise convergence of functions and convergence of functionals, Proc. Amer. Math. Soc., Volume 88 (1983) no. 3, pp. 486-490

[5] H. Brezis; L. Nirenberg Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents, Comm. Pure Appl. Math., Volume 36 (1983) no. 4, pp. 437-477

[6] J.M. Coron Topologie et cas limite des injections de Sobolev, C. R. Acad. Sci. Paris Sér. I Math., Volume 299 (1984), pp. 209-212

[7] R. Hadiji, H. Yazidi, Problem with critical Sobolev exponent and with weight, Chinese Ann. Math. Ser. B, in press

[8] R. Hadiji, R. Molle, D. Passaseo, H. Yazidi, in preparation

[9] Y. Li On $- Δ u = K (x) u^{5}$ in $R^{3}$ , Comm. Pure Appl. Math., Volume 46 (1993), pp. 303-340

[10] P.L. Lions The concentration-compactness principle in the calculus of variations, The limit case, part 1 and part 2, Rev. Mat. Iberoamericana, Volume 1 (1985) no. 1, pp. 145-201 and 1 (2) (1985) 45–121

[11] M. Musso; D. Passaseo Multibump solutions for a class of nonlinear elliptic problems, Calc. Var. Partial Differential Equations, Volume 7 (1998) no. 1, pp. 53-86

[12] O. Rey Sur un problème variationnel non compact : l'effet de petits trous dans le domaine, C. R. Acad. Sci. Paris Sér. I Math., Volume 308 (1989), pp. 349-352

[13] O. Rey The role of the Green's function in a nonlinear elliptic equation involving the critical Sobolev exponent, J. Funct. Anal., Volume 89 (1990) no. 1, pp. 1-52

[14] G. Talenti Best constants in Sobolev inequality, Ann. Mat. Pura Appl., Volume 110 (1976), pp. 353-372

[15] H. Yazidi, Etude de quelques EDP non linéaires sans compacité, Ph.D. Thesis, Université Paris 12 (2006)

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