Comptes Rendus
no. 10
Partial differential equations
Bifurcation near infinity for the Neumann problem with concave–convex nonlinearities
[Bifurcation à l'infini pour le problème de Neumann avec terme concave–convexe]

Présenté par : Philippe G. Ciarlet

Nikolaos S. Papageorgiou1 ; Vicenţiu D. Rădulescu2, 3
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
3 Simion Stoilow Institute of Mathematics of the Romanian Academy, P.O. Box 1-764, 014700 Bucharest, Romania
Comptes Rendus. Mathématique, Volume 352 (2014) no. 10, pp. 811-816.

Dans cette Note, nous étudions le problème elliptique paramétrique de Neumann pour un opérateur différentiel non homogène et avec une réaction qui présente des termes du type concave–convexe. En utilisant la condition d'Ambrosetti–Rabinowitz en combinaison avec des outils topologiques et variationnels, nous prouvons un théorème de bifurcation pour de grandes valeurs du paramètre réel.

In this Note, we study a class of Neumann parametric elliptic equations driven by a nonhomogeneous differential operator and with a reaction that exhibits competing terms (concave–convex nonlinearities). Using the Ambrosetti–Rabinowitz condition and related topological and variational arguments, we prove a bifurcation result for large values of the parameter.

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DOI : 10.1016/j.crma.2014.08.009
Nikolaos S. Papageorgiou 1 ; Vicenţiu D. Rădulescu 2, 3

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
3 Simion Stoilow Institute of Mathematics of the Romanian Academy, P.O. Box 1-764, 014700 Bucharest, Romania 
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Nikolaos S. Papageorgiou; Vicenţiu D. Rădulescu. Bifurcation near infinity for the Neumann problem with concave–convex nonlinearities. Comptes Rendus. Mathématique, Volume 352 (2014) no. 10, pp. 811-816. doi : 10.1016/j.crma.2014.08.009. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2014.08.009/

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

[4] 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

[5] S.A. Marano; N.S. Papageorgiou Positive solutions to a Dirichlet problem with p-Laplacian and concave–convex nonlinearity depending on a parameter, Commun. Pure Appl. Anal., Volume 12 (2013), pp. 815-829

[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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