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

In: Nonlinear Analysis

Author

Listed:
  • Dumitru Motreanu

    (Université de Perpignan)

  • Patrick Winkert

    (Technische Universität Berlin)

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.

Suggested Citation

  • Dumitru Motreanu & Patrick Winkert, 2012. "The Fučík Spectrum for the Negative p-Laplacian with Different Boundary Conditions," Springer Optimization and Its Applications, in: Panos M. Pardalos & Pando G. Georgiev & Hari M. Srivastava (ed.), Nonlinear Analysis, edition 127, chapter 0, pages 471-485, Springer.
    • Handle: RePEc:spr:spochp:978-1-4614-3498-6_28
    DOI: 10.1007/978-1-4614-3498-6_28
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