Abstract and Applied Analysis
Volume 2004 (2004), Issue 9, Pages 777-792
doi:10.1155/S1085337504311115
On the discreteness of the spectra of the Dirichlet and Neumann p-biharmonic problems
Centre of Applied Mathematics, University of West Bohemia, Univerzitní 22, Plzeň 306 14, Czech Republic
Received 15 August 2003
Copyright © 2004 Jiří Benedikt. 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.
Abstract
We are interested in a nonlinear boundary value problem for (|u″|p−2u″)′′=λ|u|p−2u in [0,1], p>1, with Dirichlet and Neumann boundary conditions. We prove that eigenvalues of the Dirichlet problem are positive, simple, and isolated, and form an increasing unbounded sequence. An eigenfunction, corresponding to the nth eigenvalue, has precisely n−1 zero points in (0,1). Eigenvalues of the Neumann problem are nonnegative and isolated, 0 is an eigenvalue which is not simple, and the positive eigenvalues are simple and they form an increasing unbounded sequence. An eigenfunction, corresponding to the nth positive eigenvalue, has precisely n+1 zero points in (0,1).