Periodic solutions for the coupled wave equations with concave-convex nonlinearities

In this work, we show that the coupled wave equations possess infinitely many time-periodic solutions. The nonlinearities are the combination of a convex term and a concave term, and we mainly focus on the effect of the concave term on the solution structure of the periodic-Dirichlet wave equations. The proof is based on the detailed analysis for the spectrum structure of the corresponding linear operator, and we obtain the existence and multiplicity of time-periodic solutions to our problem by critical point theory and approximation arguments.