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Forum Mathematicum

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Volume 30, Issue 3


Positive solutions for nonlinear nonhomogeneous parametric Robin problems

Nikolaos S. Papageorgiou / Vicenţiu D. RădulescuORCID iD: http://orcid.org/0000-0003-4615-5537
  • Corresponding author
  • Department of Mathematics, Faculty of Sciences, King Abdulaziz University, P.O. Box 80203, Jeddah 21589, Saudi Arabia; and Department of Mathematics, University of Craiova, Street A. I. Cuza 13, 200585 Craiova, Romania
  • orcid.org/0000-0003-4615-5537
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/ Dušan D. Repovš
Published Online: 2017-08-15 | DOI: https://doi.org/10.1515/forum-2017-0124


We study a parametric Robin problem driven by a nonlinear nonhomogeneous differential operator and with a superlinear Carathéodory reaction term. We prove a bifurcation-type theorem for small values of the parameter. Also, we show that as the parameter λ>0 approaches zero, we can find positive solutions with arbitrarily big and arbitrarily small Sobolev norm. Finally, we show that for every admissible parameter value, there is a smallest positive solution uλ* of the problem, and we investigate the properties of the map λuλ*.

Keywords: Robin boundary condition; nonlinear nonhomogeneous differential operator; nonlinear regularity; nonlinear maximum principle; bifurcation-type result; extremal positive solution

MSC 2010: 35J20; 35J60


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About the article

Received: 2017-06-12

Revised: 2017-07-13

Published Online: 2017-08-15

Published in Print: 2018-05-01

Funding Source: Javna Agencija za Raziskovalno Dejavnost RS

Award identifier / Grant number: P1-0292

Award identifier / Grant number: J1-8131

Award identifier / Grant number: J1-7025

This research was supported by the Slovenian Research Agency grants P1-0292, J1-8131, J1-7025. V. D. Rădulescu was also supported by a grant of Ministry of Research and Innovation, CNCS-UEFISCDI, project number PN-III-P4-ID-PCE-2016-0130, within PNCDI III.

Citation Information: Forum Mathematicum, Volume 30, Issue 3, Pages 553–580, ISSN (Online) 1435-5337, ISSN (Print) 0933-7741, DOI: https://doi.org/10.1515/forum-2017-0124.

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