Topological Methods in Nonlinear Analysis

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

Juha Berkovits, Herbert Leinfelder, and Vesa Mustonen

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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.

Article information

Source
Topol. Methods Nonlinear Anal., Volume 22, Number 2 (2003), 273-295.

Dates
First available in Project Euclid: 30 September 2016

Permanent link to this document
https://projecteuclid.org/euclid.tmna/1475266337

Mathematical Reviews number (MathSciNet)
MR2036377

Zentralblatt MATH identifier
1047.35100

Citation

Berkovits, Juha; Leinfelder, Herbert; Mustonen, Vesa. Existence and multiplicity results for wave equations with time-independent nonlinearity. Topol. Methods Nonlinear Anal. 22 (2003), no. 2, 273--295. https://projecteuclid.org/euclid.tmna/1475266337


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