Multiple periodic solutions of Lagrangian systems of relativistic oscillators

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].