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Transactions of the American Mathematical Society

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On a class of functionals invariant under a $ {\bf Z}\sp n$ action

Author: Paul H. Rabinowitz
Journal: Trans. Amer. Math. Soc. 310 (1988), 303-311
MSC: Primary 34C25; Secondary 35J60, 58E05, 58F22
MathSciNet review: 965755
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Abstract: Consider a system of ordinary differential equations of the form $ ({\ast})$

$\displaystyle \ddot q + {V_q}(t,\,q) = f(t)$

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.

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Keywords: $ {{\mathbf{Z}}^n}$ action periodic solution, critical point, minimax argument, Ljusternik-Schnirelmann category, Neumann problem
Article copyright: © Copyright 1988 American Mathematical Society