Bifurcation near the origin for the Robin problem with concave–convex nonlinearities

Nikolaos S. Papageorgiou; Vicenţiu D. Rădulescu

doi:10.1016/j.crma.2014.05.007

Partial differential equations

Bifurcation near the origin for the Robin problem with concave–convex nonlinearities

Presented by: Philippe G. Ciarlet

Nikolaos S. Papageorgiou ¹; Vicenţiu D. Rădulescu ²

¹ National Technical University, Department of Mathematics, Zografou Campus, Athens 15780, Greece
² Department of Mathematics, Faculty of Science, King Abdulaziz University, Jeddah, Saudi Arabia

Comptes Rendus. Mathématique, Volume 352 (2014) no. 7-8, pp. 627-632.

Abstract
Résumé

In this Note, we deal with the Robin parametric elliptic equation driven by a nonhomogeneous differential operator and with a reaction that exhibits competing terms (concave–convex nonlinearities). Without employing the Ambrosetti–Rabinowitz condition, we prove a bifurcation theorem for small positive values of the real parameter.

Dans cette Note, nous étudions le problème elliptique paramétrique de Robin pour un opérateur différentiel non homogène et avec une réaction qui présente des termes concurrents (du type concave–convexe). Sans utiliser la condition d'Ambrosetti–Rabinowitz, nous prouvons un théorème de bifurcation pour de petites valeurs positives du paramètre réel.

Received: 2014-04-04
Accepted: 2014-05-03
Published online: 2014-06-11

DOI: 10.1016/j.crma.2014.05.007

Author's affiliations:

Nikolaos S. Papageorgiou ¹; Vicenţiu D. Rădulescu ²

¹ National Technical University, Department of Mathematics, Zografou Campus, Athens 15780, Greece
² Department of Mathematics, Faculty of Science, King Abdulaziz University, Jeddah, Saudi Arabia

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     author = {Nikolaos S. Papageorgiou and Vicen\c{t}iu D. R\u{a}dulescu},
     title = {Bifurcation near the origin for the {Robin} problem with concave{\textendash}convex nonlinearities},
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Nikolaos S. Papageorgiou; Vicenţiu D. Rădulescu. Bifurcation near the origin for the Robin problem with concave–convex nonlinearities. Comptes Rendus. Mathématique, Volume 352 (2014) no. 7-8, pp. 627-632. doi : 10.1016/j.crma.2014.05.007. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2014.05.007/

References
Cited by

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

[2] A. Ambrosetti; H. Brezis; G. Cerami Combined effects of concave–convex nonlinearities in some elliptic problems, J. Funct. Anal., Volume 122 (1994), pp. 519-543

[3] H. Brezis Functional Analysis, Sobolev Spaces and Partial Differential Equations, Universitext, Springer, New York, 2011

[4] P.G. Ciarlet Linear and Nonlinear Functional Analysis with Applications, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, USA, 2013

[5] G. Lieberman The natural generalization of the conditions of Ladyzhenskaya and Uraltseva for elliptic equations, Commun. Partial Differ. Equ., Volume 16 (1991), pp. 311-361

[6] N.S. Papageorgiou, V.D. Rădulescu, Bifurcation of positive solutions for nonlinear nonhomogeneous Robin and Neumann problems with competing nonlinearities, submitted for publication.

[7] P. Pucci; J. Serrin The Maximum Principle, Birkhäuser, Basel, Switzerland, 2007

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