Bifurcation of positive solutions for nonlinear nonhomogeneous Robin and Neumann problems with competing nonlinearities

Nikolaos S. Papageorgiou; Vicenţiu D. Rădulescu

doi:10.3934/dcds.2015.35.5003

Article Contents

2015, Volume 35, Issue 10: 5003-5036. Doi: 10.3934/dcds.2015.35.5003

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Bifurcation of positive solutions for nonlinear nonhomogeneous Robin and Neumann problems with competing nonlinearities

Nikolaos S. Papageorgiou^1, and
Vicenţiu D. Rădulescu^2,

1.
Department of Mathematics, National Technical University of Athens, Zografou Campus, Athens 15780
2.
“Simion Stoilow” Mathematics Institute of the Romanian Academy, P.O. Box 1-764, 014700 Bucharest

Received: August 31, 2014

Revised: December 31, 2014

Published: September 2015

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Abstract

In this paper we deal with Robin and Neumann parametric elliptic equations driven by a nonhomogeneous differential operator and with a reaction that exhibits competing nonlinearities (concave-convex nonlinearities). For the Robin problem and without employing the Ambrosetti-Rabinowitz condition, we prove a bifurcation theorem for the positive solutions for small values of the parameter $\lambda>0$. For the Neumann problem with a different geometry and using the Ambrosetti-Rabinowitz condition we prove bifurcation for large values of $\lambda>0$.

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Mathematics Subject Classification: 35J66, 35J70, 35J92.

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