Abstract and Applied Analysis

Various Half-Eigenvalues of Scalar p -Laplacian with Indefinite Integrable Weights

Wei Li and Ping Yan

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Abstract

Consider the half-eigenvalue problem ( ϕ p ( x ) ) + λ a ( t ) ϕ p ( x + ) λ b ( t ) ϕ p ( x ) = 0 a.e. t [ 0 , 1 ] , where 1 < p < , ϕ p ( x ) = | x | p 2 x , x ± ( ) = max { ± x ( ) , 0 } for x 𝒞 0 : = C ( [ 0 , 1 ] , ) , and a ( t ) and b ( t ) are indefinite integrable weights in the Lebesgue space γ : = L γ ( [ 0 , 1 ] , ) , 1 γ . We characterize the spectra structure under periodic, antiperiodic, Dirichlet, and Neumann boundary conditions, respectively. Furthermore, all these half-eigenvalues are continuous in ( a , b ) ( γ , w γ ) 2 , where w γ denotes the weak topology in γ space. The Dirichlet and the Neumann half-eigenvalues are continuously Fréchet differentiable in ( a , b ) ( γ , γ ) 2 , where γ is the L γ norm of γ .

Article information

Source
Abstr. Appl. Anal., Volume 2009 (2009), Article ID 109757, 27 pages.

Dates
First available in Project Euclid: 16 March 2010

Permanent link to this document
https://projecteuclid.org/euclid.aaa/1268745594

Digital Object Identifier
doi:10.1155/2009/109757

Mathematical Reviews number (MathSciNet)
MR2539910

Zentralblatt MATH identifier
1188.34018

Citation

Li, Wei; Yan, Ping. Various Half-Eigenvalues of Scalar $p$ -Laplacian with Indefinite Integrable Weights. Abstr. Appl. Anal. 2009 (2009), Article ID 109757, 27 pages. doi:10.1155/2009/109757. https://projecteuclid.org/euclid.aaa/1268745594


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