Real-valued, time-periodic localized weak solutions for a semilinear wave equation with periodic potentials

Abstract

We consider the semilinear wave equation for three different classes (P1)–(P3) of periodic potentials . (P1) consists of periodically extended delta-distributions, (P2) of periodic step potentials and (P3) contains certain periodic potentials for . Among other assumptions we suppose that for some and . In each class we can find suitable potentials that give rise to a critical exponent such that for both in the ‘’ and the ‘−’ case we can use variational methods to prove existence of time-periodic 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 depends on the regularity of . 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 (considered on suitable space of time-periodic functions) is bounded away from . This 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.