The existence of solutions of a nonlinear suspension bridge equation

In this paper we investigate a relation between the multiplicity of solutions and source terms in a nonlinear suspension bridge equation in the interval $(-frac{2}{pi}, frac{2}{pi})$, under Dirichlet boundary condition $$ (0.1) u_{tt} + u_{xxxx} + bu^+ = f(x) in (-frac{2}{pi}, frac{2}{pi}) imes R, $$ $$ (0.2) u(pmfrac{2}{pi}, t) = u_{xx}(pmfrac{2}{pi}, t) = 0, $$ $$ (0.3) u is pi - periodic in t and even in x and t, $$ where the nonlinearity - $(bu^+)$ crosses an 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 $u^+$ models the fact that cables expansion but do not resist compression.