A Variational Characterization of the Fucik Spectrum and Applications

We characterize the {\it Fucik spectrum} (see \cite{fucik}) of a class selfadjoint operators. Our characterization relies on Lyapunov-Schmidt reduction arguments. We use this characterization to establish the existence of solutions for a semilinear wave equation. This work has been motivated by the authors' results in \cite{chcastro1} where one dimensional second order ordinary differential equations are studied.

A Variational Characterization of the Fucik Spectrum and Applications Una caracterización variacional del espectro de Fucik y aplicaciones ALFONSO CASTRO1, CHEN CHANG2

1Harvey Mudd College, Claremont, USA. Email: castro@math.hmc.edu 
2UTSA, San Antonio, USA. Email: chen.chang@utsa.edu 

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Abstract

We characterize the \it Fucik spectrum (see [7]) of a class selfadjoint operators. Our characterization relies on Lyapunov-Schmidt reduction arguments. We use this characterization to establish the existence of solutions for a semilinear wave equation. This work has been motivated by the authors results in [4] where one dimensional second order ordinary differential equations are studied.

Key words: Fucik spectrum, Saddle point principle, Asymptotic behavior.

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2000 Mathematics Subject Classification: 35J20, 35J25, 35J60.

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Resumen

Se caracteriza el espectro de Fucik (véase [7]) de una clase de operadores autoadjuntos. Basamos esta caracterización en el método de reducción de Lyapunov-Schmidt. Usamos esta caracterización para demostrar la existencia de soluciones a una ecuación de onda semilineal. Este trabajo ha sido motivado por los resultados de los autores en [4] donde se estudian ecuaciones diferenciales ordinarias de segundo orden.

Palabras clave: Espectro de Fucik, principio de puntos de silla, comportamiento asintótico.

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Texto completo disponible en PDF

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References

[1] A. K. Ben-Naoum, C. Fabry, and D. Smets, `Resonance with respect to the Fucik Spectrum´, Electron. J. Differential Equations, 37 (2000), 1-21.

[2] H. Brezis and L. Nirenberg, `Forced Vibrations for a Nonlinear Wave Equation´, Comm. on Pure and Applied Mathematics 31, (1978), 1-30.

[3] A. Castro, `Hammerstein Integral Equations with Indefinite Kernel´, Math. and Math. Sci. 1, (1978), 187-201.

[4] A. Castro and C. Chang, `Asymptotic Behavior of the Potential and Existence of a Periodic Solution for a Second Order Differential Equation´, Applicable Analysis 82, 11 (2003), 1029-1038.

[5] M. Cuesta and J. P. Gossez, `A Variational Approach to Nonresonance with Respect to the Fucik Spectrum´,Nonlinear Analysis T.M.A. 19, 5 (1992), 487-500.

[6] M. Cuesta, D. G. de Figueiredo, and J. P. Gossez, `The Beginning of the Fucik Spectrum for the p-Laplacian´,J. Differential Equations 159, 1 (1999), 212-238.

[7] S. Fucik, `Boundary Value Problems with Jumping Nonlinearities´, Casopis Pest. Mat. 101, (1976), 69-87.

[8] E. Massa, `On a Variational Characterization of a Part of the Fucik Spectrum and a Superlinear Equation for the Neumann p-Laplacian in Dimension One´, Adv. Differential Equations 9, 5-6 (2004a), 699-720.

[9] E. Massa, `On a Variational Characterization of the Fucik Spectrum of the Laplacian and a Superlinear Sturm-Liouville Equation´, Proc. Roy. Soc. Edinburgh Sect. A 134, 3 (2004b), 557-577.

[10] E. Massa and B. Ruf, `On the Fucik Spectrum for Elliptic Systems´, Topol. Methods Nonlinear Analysis 27, 2 (2006), 195-228.

[11] D. G. de Figueiredo and J. P. Gossez, `On the First Curve of the Fucik Spectrum of an Elliptic Operator´,Differential and Integral Equations 7, 5-6 (1994), 1285-1302.

[12] D. G. de Figueiredo and B. Ruf, `On the Periodic Fucik Spectrum and a Superlinear Sturm-Liouville Equation´,Proc. Roy. Soc. Edinburgh Sect. A. 123, 1 (1993), 95-107.

(Recibido en enero de 2009. Aceptado en abril de 2010)

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Este artículo se puede citar en LaTeX utilizando la siguiente referencia bibliográfica de BibTeX:

@ARTICLE{RCMv44n1a03, 
    AUTHOR  = {Castro, Alfonso and Chang, Chen}, 
    TITLE   = {{A Variational Characterization of the Fucik Spectrum and Applications}}, 
    JOURNAL = {Revista Colombiana de Matemáticas}, 
    YEAR    = {2010}, 
    volume  = {44}, 
    number  = {1}, 
    pages   = {23-40} 
}