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 $(0,1)$ .