Application of Local Linking to Asymptotically Linear Wave Equations with Resonance

Abstract

Existence of a time-periodic solution to a non-linear wave equation with resonance is established by a variational method. We consider the $2\pi$-periodic weak solution to a wave equation $\Box u(x,t)=h(x,t,u(x,t))$ of space dimension 1, where $h(x,t,\xi)$ is asymptotically linear in $\xi$ both as $\xi\to0$ or $\xi\to\infty$, with the co-efficient as $\xi\to\infty$ belonging to $\sigma(\Box)$. It is proved that there are some cases, where the difference of $h(t,x,\xi)$ from its linear approximation is not bounded, that guarantee the existence of a non-trivial weak solutions. The proof is based on local linking theory and $({\it WPS})^*$ condition for the existence of a non-trivial critical point of a functional.