A further multiplicity result for Lagrangian systems of relativistic oscillators

TL;DR

This paper extends previous work on Lagrangian systems of relativistic oscillators by applying minimax theorems and Brezis-Mawhin theory to establish multiple solutions for periodic boundary value problems.

Contribution

It introduces a novel combination of minimax theorems and Brezis-Mawhin theory to obtain multiple solutions for relativistic oscillator systems, advancing the mathematical understanding of such problems.

Findings

Established multiple solutions for the boundary value problem

Applied minimax theorems in the context of relativistic oscillators

Formulated a challenging conjecture related to the theory

Abstract

This is our third paper, after [4] and [5], about a joint application of the theory developed by Brezis and Mawhin in [1] with our minimax theorems ([2], [3]) to get multiple solutions of problems of the type $$\cases{(\phi(u'))'=\nabla_xF(t,u) & in $[0,T]$\cr & \cr u(0)=u(T)\ , \hskip 3pt u'(0)=u'(T)\cr}$$ which are global minima of a suitable functional over a set of Lipschitzian functions. A challenging conjecture is also formulated.