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Nonlinear nonhomogeneous singular problems

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Abstract

We consider a nonlinear Dirichlet problem driven by a nonhomogeneous differential operator with a growth of order \((p-1)\) near \(+\infty \) and with a reaction which has the competing effects of a parametric singular term and a \((p-1)\)-superlinear perturbation which does not satisfy the usual Ambrosetti–Rabinowitz condition. Using variational tools, together with suitable truncation and strong comparison techniques, we prove a “bifurcation-type” theorem that describes the set of positive solutions as the parameter \(\lambda \) moves on the positive semiaxis. We also show that for every \(\lambda >0\), the problem has a smallest positive solution \(u^*_\lambda \) and we demonstrate the monotonicity and continuity properties of the map \(\lambda \mapsto u^*_\lambda \).

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Acknowledgements

This research was supported by the Slovenian Research Agency grants P1-0292, J1-8131, N1-0114, N1-0064, and N1-0083.

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Authors and Affiliations

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

    Nikolaos S. Papageorgiou

  2. Institute of Mathematics, Physics and Mechanics, 1000, Ljubljana, Slovenia

    Nikolaos S. Papageorgiou, Vicenţiu D. Rădulescu & Dušan D. Repovš

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

    Vicenţiu D. Rădulescu

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

    Vicenţiu D. Rădulescu

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

    Dušan D. Repovš

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  1. Nikolaos S. Papageorgiou
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  2. Vicenţiu D. Rădulescu
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Correspondence to Vicenţiu D. Rădulescu.

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Communicated by M. Struwe.

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Papageorgiou, N.S., Rădulescu, V.D. & Repovš, D.D. Nonlinear nonhomogeneous singular problems. Calc. Var. 59, 9 (2020). https://doi.org/10.1007/s00526-019-1667-0

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