Existence of multiple periodic solutions for a semilinear wave equation in an $n$-dimensional ball

TL;DR

This paper proves the existence of at least three periodic solutions for a radially symmetric semilinear wave equation in an n-dimensional ball using variational methods and spectral analysis.

Contribution

It introduces a novel approach combining variational methods and saddle point reduction to establish multiple solutions in arbitrary dimensions.

Findings

At least three periodic solutions exist for the equation.

The spectral structure of the linearized problem is crucial.

A new working space construction overcomes dimensional restrictions.

Abstract

This paper is devoted to the study of periodic solutions for a radially symmetric semilinear wave equation in an $n$-dimensional ball. By combining the variational methods and saddle point reduction technique, we prove there exist at least three periodic solutions for arbitrary space dimension $n$. The structure of the spectrum of the linearized problem plays an essential role in the proof, and the construction of a suitable working space is devised to overcome the restriction of space dimension.