Abstract and Applied Analysis
Volume 2009 (2009), Article ID 109757, 27 pages
Research Article

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

Department of Mathematical Sciences, Tsinghua University, Beijing 100084, China

Received 1 May 2009; Accepted 28 June 2009

Academic Editor: Pavel Drabek

Copyright © 2009 Wei Li and Ping Yan. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


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|p2x, x±()=max{±x(), 0} for x𝒞0:=C([0,1],), and a(t) and b(t) are indefinite integrable weights in the Lebesgue space γ:=Lγ([0,1],),1γ. 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,wγ)2, where wγ 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 Lγ norm of γ.