On simplicity of spectra of \(p\)-biharmonic equations (Q1883099)

The author investigates the real eigenvalues \(\lambda\) of the following nonlinear (but still homogeneous) problem \[ (\rho(t)|u''(t)|^{p-2} u''(t))''= \lambda\chi(t)|u(t)|^{p- 2}u(t),\quad t\in [0,1],\quad Md_u(0)= 0,\quad Nd_u(1)= 0, \] where \(p> 1\), \(\rho,\chi\in C[0,1]\), \(\rho,\chi> 0\), \(M\), \(N\) are \(2\times 4\)-matrices with rank 2 and \[ d_u(t)= [|u(t)|^{p- 2}u(t),|u'(t)|^{p- 2} u'(t), \rho(t)|u''(t)|^{p- 2}u''(t), (\rho(t)|u''(t)|^{p- 2} u''(t))']^T. \] Examples with positive eigenvalues which are not simple as well as examples with negative eigenvalues are presented. On the other hand, one derives conditions which are sufficient in order that there do not exist negative eigenvalues and all eigenvalues are simple. These conditions are formulated in terms of the solutions of the equations \(Mx= 0\) and \(Ny= 0\).