The authors investigate the fourth order differential equation \[ u_{tt}+u_{xxxx}+bu^+=c\qquad x\in(0,\pi),\;t\in\mathbb R \] with free-end boundary conditions \(u_{xx}=u_{xxx}=0\) on \(\{0,\pi\}\) and \(2\pi\)-periodic boundary conditions in the \(t\)-variable. They also require the symmetry conditions \(u(x,-t)=u(x,t)\) and \(u(\pi-x,t)=u(x,t)\). Let \((\Lambda^-_n)_n\) be the sequence of negative eigenvalues of the linear problem \(u_{tt}+u_{xxxx}=\Lambda u\) with the same boundary and symmetry conditions. Using variational methods the authors prove the existence of two nontrivial solutions if \(b0\) such that there are four nontrivial solutions if \(\Lambda^-_k-\delta_k