On the existence of multiple solutions for a nonlinear wave equation
This paper is devoted to the existence of multiple time-periodic solutions for the problem \[ u_{tt}-u_{xx}=g(u)\text{ in }(0,\pi)\times {\mathbb{R}},\quad u(0,t)=u(\pi,t)=0\text{ for all } t\in {\mathbb{R}}, \] when g is continuous. The author shows that the number of weak solutions is not smaller than the number of those eigenvalues of \(\partial^ 2_ t-\partial^ 2_ x\) corresponding to the odd eigenfunctions and which are crossed by \(g(\xi)/\xi\) as \(\xi\) varies from 0 to \(\infty\). The proof is based on an existence theorem for a pseudo- monotone mapping.
scientific article; zbMATH DE number 4059967