Nontrivial solutions for a floating beam equation (Q1809066)

The authors consider a ship modelled as a floating beam, with the following beam equation \[ \begin{cases} u_{tt}(x,t)+ u_{xxxx}(x, t)+ b[u(x, t)]^+= c\quad\text{in }[0,\pi]\times \mathbb{R},\\ u_{xx}(0, t)= u_{xx}(\pi, t)= u_{xxx}(0,t)= u_{xxx}(\pi, t)= 0\quad\forall t\in\mathbb{R},\\ u(x,t)= u(x, t+ 2\pi),\\ u(x,t)= u(x,-t),\\ u(x,t)= u(\pi- x,t)\end{cases} \] where \([u(x,t)]^+= \max \{0, u(x,t)\}\). By using a variational approach a theorem of existence of at least two nontrivial solutions is given.