Multiple periodic solutions of Lagrangian systems of relativistic oscillators

TL;DR

This paper establishes conditions for the existence of multiple periodic solutions in relativistic Lagrangian systems, extending previous results that guaranteed only a single solution, using advanced variational methods.

Contribution

It introduces new conditions ensuring at least two global minima of the associated functional, demonstrating the multiplicity of solutions in relativistic oscillators.

Findings

Proves existence of at least two solutions under specified conditions.

Extends prior single-solution results to multiple solutions.

Utilizes advanced variational techniques for proof.

Abstract

Let $B_L$ the open ball in ${\bf R}^n$ centered at $0$, of radius $L$, and let $\phi$ be a homeomorphism from $B_L$ onto ${\bf R}^n$ such that $\phi(0)=0$ and $\phi=\nabla\Phi$, where the function $\Phi:\bar {B_L}\to ]-\infty,0]$ is continuous and strictly convex in $\bar {B_L}$, and of class $C^1$ in $B_L$. Moreover, let $F:[0,T]\times {\bf R}^n\to {\bf R}$ be a function which is measurable in $[0,T]$, of class $C^1$ in ${\bf R}^n$ and such that $\nabla_xF$ satisfies the $L^1$-Carath\'eodory conditions. Set $$K=\{u\in Lip([0,T],{\bf R}^n) : |u'(t)|\leq L\ for\ a.e.\ t\in [0,T] , u(0)=u(T)\}\ ,$$ and define the functional $I:K\to {\bf R}$ by $$I(u)=\int_0^T(\Phi(u'(t))+F(t,u(t)))dt$$ for all $u\in K$. In [1], Brezis and Mawhin proved that any global minimum of $I$ in $K$ is a solution of the problem $$\cases{(\phi(u'))'=\nabla_xF(t,u) & in $[0,T]$\cr & \cr u(0)=u(T)\ , u'(0)=u'(T)\ .\cr}$$ In the present paper, we provide a set of conditions under which the functional $I$ has at least two global minima in $K$. This seems to be the first result of this kind. The main tool of our proof is the well-posedness result obtained in [3].