Real-valued, time-periodicweak solutions for a semilinear wave equation with periodic δ-potential

We consider the semilinear wave equation $V(x)$$u$$_{u}$ − $u$$_{xx}$ = $±|u|$$^{p-1}$$u$ with p ∈ (1, $\frac{5}{3}$) and a periodically extended delta potential $V(x)$ = $α + βδper(x)$. Both the “+” and the “-” case can be treated. We prove the existence of time-periodic real-valued solutions that are localized in the space direction. Our result builds upon a Fourier-Floquet-Bloch expansion of the solution and a detailed analysis of the spectrum of the wave operator. In fact, it turns out that by a careful choice of the parameters α, β and the spatial and temporal periods, the spectrum of the wave operator $V(x)$∂$\frac{2}{t}$ - ∂$\frac{2}{x}$ (considered on suitable space of time-periodic functions) is bounded away from 0. This allows to find weak solutions as critical points of a functional on a suitable Hilbert space and to apply tools for indefinite variational problems.