A nonlinear wave equation with jumping nonlinearity (Q1864063)

The authors study multiplicity of solutions \(u(x,t)\) for a piecewise linear perturbation of the one-dimensional wave operator \(u_{tt}-u_{xx}\) under Dirichlet boundary condition on the interval \((-{\pi\over 2}, {\pi\over 2})\) and periodic condition on \(t\), that is \[ \begin{cases} u_{tt}-u_{xx}+g (u)=f(x,t) \text{ in }\left(-{\pi\over 2}, {\pi\over 2}\right)\times \mathbb{R},\\ u \left(\pm {\pi\over 2},t\right)= 0,\\ u\text{ is }\pi\text{-periodic in }t\text{ and even in }x\text{ and }t,\end{cases} \] where the authors assume that \(g(u)=b u^+-au^-\). The goal of the authors is to investigate a relation between multiplicity of solutions and source terms, when the nonlinearity \(-(bu^+-au^-)\) crosses two eigenvalues and the source term \(f\) is generated by two eigenfunctions \(\varphi_{0,0}\), \(\varphi_{1,0}\).