  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.
