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

Abstract

We are interested in a nonlinear boundary value problem for ${\left({\left|u^{″}\right|}^{p-2}{u}^{″}\right)}^{\prime \text{}\prime }=\lambda {\left|u\right|}^{p-2}u$ in $\left[0,1\right]$, $p>1$, 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 $\left(0,1\right)$. 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 $\left(0,1\right)$.