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

DOI: http://dx.doi.org/10.12775/TMNA.2003.040

#### 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.

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.

#### Keywords

Wave equation; multiple solutions; degree theory; free vibrations

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