Existence and multiplicity results for wave equations with time-independent nonlinearity

Abstract

We shall study the existence of time-periodic solutions for a semilinear
wave equation with a given time-independent nonlinear perturbation and
small forcing. Since the distribution of eigenvalues of the linear part
varies with the period, the solvability of the problem depends essentially
on the frequency. The main idea of this paper is to consider the situation
where the period is not prescribed and hence treated as a parameter.
The description of the distribution of eigenvalues as a function of
the period enables us to show that under certain conditions the
interaction between the nonlinearity and the spectrum of the wave
operator induces multiple solutions.
Our basic new result states that the autonomous equation admits
at least two nontrivial solutions (free vibrations) for a restricted
(but infinite) set of periods such that the nonlinearity interacts with
one simple eigenvalue. As a corollary we prove that the semilinear wave
equation with time-independent nonlinearity and small forcing admits an
infinite sequence of pairs of periodic solutions with corresponding period
tending to zero. The results are obtained via generalized topological
degree theory.