Free vibrations for an asymmetric beam equation (Q698852)

The paper shows the existence of a nontrivial \(2\pi\)-periodic weak solution of the following beam equation: \[ \begin{gathered}\frac{\partial ^2u}{\partial t^2} + \frac{\partial ^4u}{\partial x^4}+g(u)=0 ,\\ u(0,t)=u(\pi, t)=\frac{\partial ^2u}{\partial x^2}(0,t)=\frac{\partial ^2u}{\partial x^2}(\pi ,t)=0 , \end{gathered} \] where \(g\) has the form \[ g(s)=\begin{cases} |s|^{p-2}s,\quad s\geq 0 \\ |s|^{q-2}s,\quad s<0 , \end{cases} \] where \(p,q>2\) and \(p\neq q\). However, the result of the paper holds for more general cases. The proof is based on the Galerkin approximation method along with a saddle point variational method.