Free vibrations of beams on bearings with nonlinear elastic responses (Q1288862)

The existence of nontrivial time periodic solutions to the problem \[ u_{tt}+ u_{xxxx}= 0, \] \[ u_{xx}(0,\cdot)= u_{xx}\Biggl({\pi\over 4},\cdot\Biggr)= 0,\tag{1} \] \[ u_{xxx}(0,\cdot)= -f(u(0,\cdot)),\quad u_{xxx}\Biggl({\pi\over 4},\cdot\Biggr)= f\Biggl(u\Biggl({\pi\over 4},\cdot\Biggr)\Biggr), \] under some conditions on \(f\), is proved by using Morse indices. A semilinear version, when (1) is replaced by the equation \(u_{tt}+ u_{xxxx}+ g(u)= 0\), is also studied.