## Abstract and Applied Analysis

### On the discreteness of the spectra of the Dirichlet and Neumann $p$-biharmonic problems

Jiří Benedikt

#### Abstract

We are interested in a nonlinear boundary value problem for $(|u''|^{p-2}u'')''=\lambda|u|^{p-2}u$ 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 $n$th 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 $n$th positive eigenvalue, has precisely $n+1$ zero points in $(0,1)$.

#### Article information

Source
Abstr. Appl. Anal., Volume 2004, Number 9 (2004), 777-792.

Dates
First available in Project Euclid: 10 October 2004

https://projecteuclid.org/euclid.aaa/1097458587

Digital Object Identifier
doi:10.1155/S1085337504311115

Mathematical Reviews number (MathSciNet)
MR2102601

Zentralblatt MATH identifier
1081.34018

Subjects
Primary: 34B15: Nonlinear boundary value problems 34C10
Secondary: 47J10

#### Citation

Benedikt, Jiří. On the discreteness of the spectra of the Dirichlet and Neumann $p$-biharmonic problems. Abstr. Appl. Anal. 2004 (2004), no. 9, 777--792. doi:10.1155/S1085337504311115. https://projecteuclid.org/euclid.aaa/1097458587