On the discreteness of the spectra of the Dirichlet and Neumannp-biharmonic problems

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)