Multiple periodic solutions of Lagrangian systems of relativistic oscillators

Let $B_L$ the open ball in ${\mathbf{R}}^n$ centered at $0$, of radius $L$, and let $\varphi$ be a homeomorphism from $B_L$ onto ${\mathbf{R}}^n$ such that $\varphi (0)=0$ and $\varphi =\mathrm{\nabla }\mathrm{\Phi }$, where the function $\mathrm{\Phi }:\overline{B_L}\to ]-\mathrm{\infty },0]$ is continuous and strictly convex in $\overline{B_L}$, and of class $C^1$ in $B_L$. Moreover, let $F:[0,T]\times {\mathbf{R}}^n\to \mathbf{R}$ be a function which is measurable in $[0,T]$, of class $C^1$ in ${\mathbf{R}}^n$ and such that ${\mathrm{\nabla }}_{x}F$ satisfies the $L^1$-Carath'eodory conditions. Set $$K=\{u\in Lip([0,T],{\mathbf{R}}^n):\mathrm{\mid }{u}^{\mathrm{\prime }}(t)\mathrm{\mid }\le Lfora\mathrm{.}e\mathrm{.}t\in [0,T],u(0)=u(T)\},$$ and define the functional $I:K\to \mathbf{R}$ by $$I(u)={\int }_0^T(\mathrm{\Phi }(u^{\mathrm{\prime }}(t))+F(t,u(t)))dt$$ 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].