Real-valued, time-periodic localizedweak solutions for a semilinearwave equation with periodic potentials

We consider the semilinear wave equation V(x)u$_{tt}$ − uₓₓ + q(x)u = ±f (x, u) for three different classes (P1), (P2), (P3) of periodic potentials V, q. (P1) consists of periodically extended deltadistributions, (P2) of periodic step potentials and (P3) contains certain periodic potentials V, q ∈ Hr per(R) for r ∈ [1, 3/2). Among other assumptions we suppose that | f (x, s)| ≤ c(1 + |s|p) for some c > 0 and p > 1. In each class we can find suitable potentials that give rise to a critical exponent p∗ such that for p ∈ (1, p∗) both in the “+” and the “-” case we can use variational methods to prove existence of timeperiodic real-valued solutions that are localized in the space direction. The potentials are constructed explicitely in class (P1) and (P2) and are found by a recent result from inverse spectral theory in class (P3). The critical exponent p∗ depends on the regularity of V, q. Our result builds upon a Fourier 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 potentials and the spatial and temporal periods, the spectrum of the wave operator V(x)∂$\substack{2\\t}$− ∂$\substack{2\\x}$+ q(x) (considered on suitable space of time-periodic functions) is bounded away from 0. ... mehrThis allows to find weak solutions as critical points of a functional on a suitable Hilbert space and to apply tools for strongly indefinite variational problems.