Response solutions for elliptic-hyperbolic equations with nonlinearities and periodic external forces

In this paper, we focus on the existence of response solutions, i.e. periodic solutions with the same frequencies as the external forces, for elliptic-hyperbolic partial differential equations with nonlinearities and periodic forces. The main tools are Lyapunov–Schmidt reduction and Nash–Moser iteration scheme, both of which have demonstrated success in hyperbolic scenarios. At each step of the iteration, the Galerkin approximation of the equation is solved. The new issue is that the spectral theory of the generalized Sturm–Liouville problem is employed, which also introduces new difficulties for estimations at each step. Under appropriate non-resonance conditions on the frequency, the existence of response solutions for the model will be established.