On the Asymmetric Beam Equation

We investigate the nonlinear beam equation in an interval $(\pi/2;\pi/2)$, with Dirichlet Boundary Condition,
\begin{equation} \label{p1} u_{tt} + u_{xxxx} + bu^{+} = f(x,t), \qquad \text{in} \qquad (\pi/2;\pi/2) \times R \end{equation}
$u$ is periodic in $t$ and even in $x$ and $t$; here the nonlinearity $(bu^{+})$ crosses the eigenvalue $\lambda_{10}$. This equation represents a bending beam supported by cables under a load $f$.
The constant $b$ represents the restoring force if the cables stretch. The nonlinearity $(bu^{+})$ models the fact that cables resist expansion but do not resist compression. McKenna and Walter [1] showed by degree theory that equation \eqref{p1} with constant load $1+h$ ($h$ is bounded) has at least two solutions.
This is a joint work with Tacksun Jung (Kunsan National University, Kunsan, South Korea).