On the discreteness of the spectra of the Dirichlet and Neumann \(p\)-biharmonic problems
The author studies the structure of eigenvalues and eigenfunctions of the nonlinear boundary value problems for \[ (| u^{\prime \prime }(t)| ^{p-2}u^{\prime \prime }(t))^{\prime \prime }=\lambda | u(t)| ^{p-2}u(t),\quad t\in [ 0,1],\;p>1,\text{ and }\lambda >1, \] with Dirichlet and Neumann boundary conditions. This work is a continuation of the one done by \textit{P. Drábek} and \textit{M. Ôtani} [Electron. J. Differ. Equ. 2001, Paper No. 48, 19 p., electronic only (2001; Zbl 0983.35099)]\ which deals with the Navier boundary value problem. For the Dirichlet case, he proves that eigenvalues are positive, simple, isolated and form an increasing unbounded sequence. An eigenfunction, corresponding to the \(n\)th eigenvalue, has precisely \(n-1\) zeros in \( (0,1).\) For the Neumann case, the author proves that positive eigenvalues are isolated and simple. An eigenfunction, corresponding to the \(n\)th eigenvalue, has precisely \(n+1\) zeros in \((0,1).\) \(0\) is also an isolated eigenvalue but not simple. Finally, he establishes a relation between the \(n\)th positive eigenvalues of Dirichlet problem and Neumann problem and he cited some open problems.
scientific article; zbMATH DE number 2163687
scientific article