Nodal solutions for scalar curvature type equations with perturbation terms on compact Riemannian manifolds
Djadli, Zindine ; Jourdain, Antoinette
Bollettino dell'Unione Matematica Italiana, Tome 5-A (2002), p. 205-226 / Harvested from Biblioteca Digitale Italiana di Matematica
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In this paper we study the nodal solutions for scalar curvature type equations with perturbation. The main results concern the existence of such solutions and the exact description of their zero set. From this we deduce, in particular cases, some multiplicity results.

L'oggetto del presente articolo è lo studio delle soluzioni soggette a cambiamenti di segno delle equazioni di tipo curvatura scalare a perturbazione. I principali risultati in esso contenuti riguardano l'esistenza di tali soluzioni e la determinazione puntuale del loro insieme degli zeri. Da ciò deduciamo, in alcuni casi, dei risultati di molteplicità.

Publié le : 2002-02-01
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     author = {Zindine Djadli and Antoinette Jourdain},
     title = {Nodal solutions for scalar curvature type equations with perturbation terms on compact Riemannian manifolds},
     journal = {Bollettino dell'Unione Matematica Italiana},
     volume = {5-A},
     year = {2002},
     pages = {205-226},
     zbl = {1177.58016},
     mrnumber = {1881932},
     language = {en},
     url = {http://dml.mathdoc.fr/item/BUMI_2002_8_5B_1_205_0}
}

Djadli, Zindine; Jourdain, Antoinette. Nodal solutions for scalar curvature type equations with perturbation terms on compact Riemannian manifolds. Bollettino dell'Unione Matematica Italiana, Tome 5-A (2002) pp. 205-226. http://gdmltest.u-ga.fr/item/BUMI_2002_8_5B_1_205_0/

[1] Ambrosetti, A.-Rabinowitz, P., Dual variational methods in critical point theory and applications, Journal of Functional Analysis, 14 (1973), 349-381. | MR 370183 | Zbl 0273.49063

[2] Atkinson, F. V., Brézis, H. - Peletier, L. A., Nodal solutions of elliptic equations with critical Sobolev exponents, Journal of Differential Equations, 85 (1990), 151-170. | MR 1052332 | Zbl 0702.35099

[3] Aubin, T., Equations différentielles non linéaires et problème de Yamabe concernant la courbure scalaire, Journal de Mathématiques Pures et Appliquées, 55 (1976), 269-296. | MR 431287 | Zbl 0336.53033

[4] Brézis, H.-Nirenberg, L., Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents, Communications on Pure and Applied Mathematics, 36 (1983), 437-477. | MR 709644 | Zbl 0541.35029

[5] Brézis, H.-Nirenberg, L., Remarks on finding critical points, Communications on Pure and Applied Mathematics, 44 (1991), 939-963. | MR 1127041 | Zbl 0751.58006

[6] Cao, D.-Noussair, E. S., Multiple positive and nodal solutions for semilinear elliptic problems with critical exponents, Indiana University Mathematics Journal, 44 (1995), 1249-1271. | MR 1386768 | Zbl 0849.35030

[7] Demengel, F.-Hebey, E., On some nonlinear equations involving the p-laplacian with critical Sobolev growth and perturbation terms, to appear in Applicable Analysis. | MR 1775436 | Zbl 1005.35047

[8] Djadli, Z., Equations de Yamabe perturbées, Thèse de l'Université de Cergy-Pontoise, 1998.

[9] Djadli, Z., Nonlinear elliptic equations with critical Sobolev exponent on compact riemannian manifolds, Calculus of Variations and Partial Differential Equations, 8 (1999), 293-326. | MR 1700267 | Zbl 0953.58017

[10] Fortunato, D.-Jannelli, E., Infinitely many solutions for some nonlinear elliptic problems in symmetrical domains, Proceedings of the Royal Society of Edinburgh Sect. A, 105 (1987), 205-213. | MR 890056 | Zbl 0676.35024

[11] Hebey, E., La méthode d'isométries concentration dans le cas d'un problème non linéaire sur les variétés compactes à bord avec exposant critique de Sobolev, Bulletin des Sciences Mathématiques, 116 (1992), 36-51. | MR 1154371 | Zbl 0756.35028

[12] Hebey, E., Introduction à l'analyse non linéaire sur les variétés, Diderot Editions, Collection Fondations, 1997. | Zbl 0918.58001

[13] Hebey, E.-Vaugon, M., Existence and multiplicity of nodal solutions for nonlinear elliptic equations with critical Sobolev growth, Journal of Functional Analysis, 119 (1994), 298-318. | MR 1261094 | Zbl 0798.35052

[14] Jourdain, A., Solutions nodales pour des équations du type courbure scalaire sur la sphère, Bulletin des Sciences Mathématiques, 123 (1999), 299-327. | MR 1697459 | Zbl 1126.53309

[15] Kazdan, J. L.-Warner, F. W., Remarks on some quasilinear elliptic equations, Communications on Pure and Applied Mathematics, 28 (1975), 567-597. | MR 477445 | Zbl 0325.35038

[16] Musso, M.-Passaseo, D., Sign changing solutions of nonlinear elliptic equations, Advances in Differential Equations, 1 (1996), 1025-1052. | MR 1409898 | Zbl 0864.35044

[17] Tarantello, G., Nodal solutions of semilinear elliptic equations with critical exponent, Differential Integral Equations, 5 (1992), 25-42. | MR 1141725 | Zbl 0758.35035

[18] Trudinger, N., Remarks concerning the conformal deformation of Riemannian structures on compact manifolds, Annali della Scuola Normale Superiore di Pisa, 22 (1968), 265-274. | MR 240748 | Zbl 0159.23801

[19] Yamabe, H., On a deformation of Riemannian structures on compact manifolds, Osaka Mathematical Journal, 12 (1960), 21-37. | MR 125546 | Zbl 0096.37201