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Nonlinear Nonhomogeneous Boundary Value Problems with Competition Phenomena

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Abstract

We consider a nonlinear boundary value problem driven by a nonhomogeneous differential operator. The problem exhibits competing nonlinearities with a superlinear (convex) contribution coming from the reaction term and a sublinear (concave) contribution coming from the parametric boundary (source) term. We show that for all small parameter values \(\lambda >0\), the problem has at least five nontrivial smooth solutions, four of constant sign and one nodal. We also produce extremal constant sign solutions and determine their monotonicity and continuity properties as the parameter \(\lambda >0\) varies. In the semilinear case we produce a sixth nontrivial solution but without any sign information. Our approach uses variational methods together with truncation and perturbation techniques, and Morse theory.

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Acknowledgements

The authors wish to thank the two knowledgeable referees for their comments, remarks and constructive criticism. This research was supported by the Slovenian Research Agency Grants P1-0292, J1-8131, J1-7025, and N1-0064.

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Authors and Affiliations

  1. Department of Mathematics Zografou Campus, National Technical University, 15780, Athens, Greece

    Nikolaos S. Papageorgiou

  2. Faculty of Applied Mathematics, AGH University of Science and Technology, al. Mickiewicza 30, 30-059, Kraków, Poland

    Vicenţiu D. Rădulescu

  3. Department of Mathematics, University of Craiova, 200585, Craiova, Romania

    Vicenţiu D. Rădulescu

  4. Faculty of Education and Faculty of Mathematics and Physics, University of Ljubljana, 1000, Ljubljana, Slovenia

    Dušan D. Repovš

Authors
  1. Nikolaos S. Papageorgiou
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  2. Vicenţiu D. Rădulescu
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  3. Dušan D. Repovš
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Correspondence to Vicenţiu D. Rădulescu.

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Papageorgiou, N.S., Rădulescu, V.D. & Repovš, D.D. Nonlinear Nonhomogeneous Boundary Value Problems with Competition Phenomena. Appl Math Optim 80, 251–298 (2019). https://doi.org/10.1007/s00245-017-9465-6

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