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

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- by Paul H. Rabinowitz
- Trans. Amer. Math. Soc.
**310**(1988), 303-311 - DOI: https://doi.org/10.1090/S0002-9947-1988-0965755-5
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## 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.## References

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## Bibliographic Information

- © Copyright 1988 American Mathematical Society
- Journal: Trans. Amer. Math. Soc.
**310**(1988), 303-311 - MSC: Primary 34C25; Secondary 35J60, 58E05, 58F22
- DOI: https://doi.org/10.1090/S0002-9947-1988-0965755-5
- MathSciNet review: 965755