On a class of functionals invariant under a 𝑍ⁿ action

Abstract

Consider a system of ordinary differential equations of the form ( ∗ ) ({\ast }) \[ q ¨ + V q ( t , q ) = f ( t ) \ddot q + {V_q}(t,\,q) = f(t) \] where f f and V V are periodic in t t , V V is periodic in the components of q = ( q 1 , … , q n ) q = ({q_1}, \ldots ,{q_n}) , and the mean value of f f vanishes. By showing that a corresponding functional is invariant under a natural Z n {{\mathbf {Z}}^n} action, a simple variational argument yields at least n + 1 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.