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The Fučík Spectrum for the Negative p-Laplacian with Different Boundary Conditions

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Nonlinear Analysis

Part of the book series: Springer Optimization and Its Applications ((SOIA,volume 68))

Abstract

This chapter represents a survey on the Fučík spectrum of the negative p-Laplacian with different boundary conditions (Dirichlet, Neumann, Steklov, and Robin). The close relationship between the Fučík spectrum and the ordinary spectrum is briefly discussed. It is also pointed out that for every boundary condition there exists a first nontrivial curve in the Fučík spectrum which has important properties such as Lipschitz continuity, being decreasing and a certain asymptotic behavior depending on the boundary condition. As a consequence, one obtains a variational characterization of the second eigenvalue λ 2 of the negative p-Laplacian with the corresponding boundary condition. The applicability of the abstract results is illustrated to elliptic boundary value problems with jumping nonlinearities.

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Notes

  1. 1.

    Svatopluk Fučík (21st October 1944 – 18th May 1979) was a Czech mathematician.

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Authors and Affiliations

  1. Département de Mathématiques, Université de Perpignan, Avenue Paul Alduy 52, 66860, Perpignan Cedex, France

    Dumitru Motreanu

  2. Institut für Mathematik, Technische Universität Berlin, Straße des 17. Juni 136, 10623, Berlin, Germany

    Patrick Winkert

Authors
  1. Dumitru Motreanu
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  2. Patrick Winkert
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Correspondence to Dumitru Motreanu .

Editor information

Editors and Affiliations

  1. , Department of Industrial & Systems Engin, University of Florida, Weil Hall 401, Gainesville, 32611, Florida, USA

    Panos M. Pardalos

  2. Center for Applied Optimization, Dept. Industrial & Systems, University of Florida, Weil Hall 303, Gainesville, 32611-6595, Florida, USA

    Pando G. Georgiev

  3. , Department of Mathematics & Statistics, University of Victoria, Finnerty Road 3800, Victoria, V8W 3R4, British Columbia, Canada

    Hari M. Srivastava

Additional information

Dedicated to Professor Themistocles M. Rassias on the occasion of his 60th birthday.

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Motreanu, D., Winkert, P. (2012). The Fučík Spectrum for the Negative p-Laplacian with Different Boundary Conditions. In: Pardalos, P., Georgiev, P., Srivastava, H. (eds) Nonlinear Analysis. Springer Optimization and Its Applications, vol 68. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-3498-6_28

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