On a class of functionals invariant under a $\textbf {Z}^ n$ action

Abstract:

Consider a system of ordinary differential equations of the form $({\ast })$ \[ \ddot q + {V_q}(t, q) = f(t)\] where $f$ and $V$ are periodic in $t$, $V$ is periodic in the components of $q = ({q_1}, \ldots ,{q_n})$, and the mean value of $f$ vanishes. By showing that a corresponding functional is invariant under a natural ${{\mathbf {Z}}^n}$ action, a simple variational argument yields at least $n + 1$ distinct periodic solutions of (*). More general versions of (*) are also treated as is a class of Neumann problems for semilinear elliptic partial differential equations.