Various Half-Eigenvalues of Scalar p-Laplacian with Indefinite Integrable Weights
Consider the half-eigenvalue problem (ϕp(x′))′+λa(t)ϕp(x+)−λb(t)ϕp(x−)=0 a.e. t∈[0,1], where 1<p<∞, ϕp(x)=|x|p−2x, x±(⋅)=max{±x(⋅), 0} for x∈𝒞0:=C([0,1],ℝ), and a(t) and b(t) are indefinite integrable weights in the Lebesgue space ℒγ:=Lγ([0,1],ℝ),1≤γ≤∞. We characterize the spectr...
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doaj-7b2c015f8c6b4467bf2bf5a3ffa2e3722020-11-24T22:07:40ZengHindawi LimitedAbstract and Applied Analysis1085-33751687-04092009-01-01200910.1155/2009/109757109757Various Half-Eigenvalues of Scalar p-Laplacian with Indefinite Integrable WeightsWei Li0Ping Yan1Department of Mathematical Sciences, Tsinghua University, Beijing 100084, ChinaDepartment of Mathematical Sciences, Tsinghua University, Beijing 100084, ChinaConsider the half-eigenvalue problem (ϕp(x′))′+λa(t)ϕp(x+)−λb(t)ϕp(x−)=0 a.e. t∈[0,1], where 1<p<∞, ϕp(x)=|x|p−2x, x±(⋅)=max{±x(⋅), 0} for x∈𝒞0:=C([0,1],ℝ), and a(t) and b(t) are indefinite integrable weights in the Lebesgue space ℒγ:=Lγ([0,1],ℝ),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)∈(ℒγ,wγ)2, where wγ denotes the weak topology in ℒγ space. The Dirichlet and the Neumann half-eigenvalues are continuously Fréchet differentiable in (a,b)∈(ℒγ,‖⋅‖γ)2, where ‖⋅‖γ is the Lγ norm of ℒγ.http://dx.doi.org/10.1155/2009/109757 |
collection |
DOAJ |
language |
English |
format |
Article |
sources |
DOAJ |
author |
Wei Li Ping Yan |
spellingShingle |
Wei Li Ping Yan Various Half-Eigenvalues of Scalar p-Laplacian with Indefinite Integrable Weights Abstract and Applied Analysis |
author_facet |
Wei Li Ping Yan |
author_sort |
Wei Li |
title |
Various Half-Eigenvalues of Scalar p-Laplacian with Indefinite Integrable Weights |
title_short |
Various Half-Eigenvalues of Scalar p-Laplacian with Indefinite Integrable Weights |
title_full |
Various Half-Eigenvalues of Scalar p-Laplacian with Indefinite Integrable Weights |
title_fullStr |
Various Half-Eigenvalues of Scalar p-Laplacian with Indefinite Integrable Weights |
title_full_unstemmed |
Various Half-Eigenvalues of Scalar p-Laplacian with Indefinite Integrable Weights |
title_sort |
various half-eigenvalues of scalar p-laplacian with indefinite integrable weights |
publisher |
Hindawi Limited |
series |
Abstract and Applied Analysis |
issn |
1085-3375 1687-0409 |
publishDate |
2009-01-01 |
description |
Consider the half-eigenvalue problem (ϕp(x′))′+λa(t)ϕp(x+)−λb(t)ϕp(x−)=0 a.e.
t∈[0,1], where 1<p<∞, ϕp(x)=|x|p−2x, x±(⋅)=max{±x(⋅), 0} for x∈𝒞0:=C([0,1],ℝ), and a(t) and b(t) are indefinite integrable weights in the Lebesgue
space ℒγ:=Lγ([0,1],ℝ),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)∈(ℒγ,wγ)2, where wγ denotes the weak topology in ℒγ space. The Dirichlet and the Neumann half-eigenvalues are continuously Fréchet differentiable in (a,b)∈(ℒγ,‖⋅‖γ)2, where ‖⋅‖γ is the Lγ norm of ℒγ. |
url |
http://dx.doi.org/10.1155/2009/109757 |
work_keys_str_mv |
AT weili varioushalfeigenvaluesofscalarplaplacianwithindefiniteintegrableweights AT pingyan varioushalfeigenvaluesofscalarplaplacianwithindefiniteintegrableweights |
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