Existence of infinitely many small periodic solutions for a semilinear variable coefficients wave equation with resonant potential

This paper is devoted to the study of periodic solutions to a semilinear variable coefficients wave equation with resonance potential under the periodic boundary conditions. This mathematical model is used to account for the forced vibrations of a nonhomogeneous string and the propagation of seismic waves in nonisotropic media. We characterize some properties of the spectrum of the variable coefficients wave operator for the periodic boundary conditions, especially we prove the existence of the essential spectral point. Under some suitable assumptions on the resonant potential, by combining variational methods and the Galerkin approximation argument, we get the infinitely many small periodic solutions of the resonant problem. Our results contain the case of zero potential and can be applied to the corresponding classical one dimensional wave equation.