Relative Morse index theory and applications in wave equations

TL;DR

This paper develops a new relative Morse index theory for linear self-adjoint operators without compactness assumptions, and applies it to establish existence and multiplicity results for periodic solutions of wave equations.

Contribution

It introduces a generalized relative Morse index theory for self-adjoint operators and extends saddle point reduction methods for wave equation solutions.

Findings

Established a relationship between different index definitions.

Proved existence of multiple periodic solutions for wave equations.

Extended critical point theories using the new index framework.

Abstract

We develop the relative Morse index theory for linear self-adjoint operator equation without compactness assumption and give the relationship between the index defined in [44] and [45]. Then we generalize the method of saddle point reduction and get some critical point theories by the index, topology degree and critical point theory. As applications, we consider the existence and multiplicity of periodic solutions of wave equations.