Quasi-periodic Solutions of Completely Resonant Wave Equations with Quasi-periodically forced Vibrations

In this paper, we provide the existence of quasi-periodic solutions with two frequencies for a class of completely resonant nonlinear wave equations with quasi-periodically forced vibrations under the spatial periodic boundary conditions. We consider the frequencies vector (ω1,ω2) near the linear system. The proofs are based on the Variational Lyapunov-Schmidt reduction and Linking Theorem.