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Sign-Changing Subharmonic Solutions to Unforced Equations with Singular ϕ-Laplacian

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Differential and Difference Equations with Applications

Part of the book series: Springer Proceedings in Mathematics & Statistics ((PROMS,volume 47))

Abstract

We prove the existence of infinitely many subharmonic solutions (with a precise nodal characterization) to the equation

$$\displaystyle{\Big( \frac{u^{\prime}} {\sqrt{1 - {u^{\prime}}^{2}}}\Big)^{\prime} + g(t,u) = 0,}$$

in the unforced case g(t,0) ≡ 0. The proof is performed via the Poincaré–Birkhoff fixed point theorem.

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Notes

  1. 1.

    Incidentally, observe that this situation is really different from the linear problem \(u^{\prime\prime} + \lambda u = 0\); in particular, here resonance phenomena do not appear for any λ > 0 (see [1, Remark 6]).

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Acknowledgements

The authors wish to thank SISSA for the financial support which has given the pleasant opportunity of taking part in the International Conference on Differential and Difference Equations and Applications in Ponta Delgada, July 2011.

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Authors and Affiliations

  1. Dipartimento di Matematica, Università di Torino, via Carlo Alberto 10, 10123, Torino, Italy

    Alberto Boscaggin

  2. Dipartimento di Scienze Matematiche, Università Politecnica delle Marche, aia Brecce Bianche, Ancona, Italy

    Maurizio Garrione

Authors
  1. Alberto Boscaggin
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  2. Maurizio Garrione
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Corresponding author

Correspondence to Alberto Boscaggin .

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Editors and Affiliations

  1. Departamento de Ciências Exactas e Naturais, Academia Militar, Amadora, Portugal

    Sandra Pinelas

  2. University of Zürich Institute of Mathematics, Zürich, Switzerland

    Michel Chipot

  3. Masaryk University Dept. Mathematics, Brno, Czech Republic

    Zuzana Dosla

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Boscaggin, A., Garrione, M. (2013). Sign-Changing Subharmonic Solutions to Unforced Equations with Singular ϕ-Laplacian. In: Pinelas, S., Chipot, M., Dosla, Z. (eds) Differential and Difference Equations with Applications. Springer Proceedings in Mathematics & Statistics, vol 47. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-7333-6_25

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