Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



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
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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.

References [Enhancements On Off] (What's this?)

  • [1] J. Mawhin and M. Willem, Multiple solutions of the periodic boundary value problem for some forced pendulum-type equations, J. Differential Equations 52 (1984), 264-287. MR 741271 (85h:34050)
  • [2] P. Pucci and J. Serrin, A mountain pass theorem, J. Differential Equations 60 (1985), 142-149. MR 808262 (86m:58038)
  • [3] -, Extensions of the mountain pass theorem, Univ. of Minnesota Math. Rep. 83-150.
  • [4] Shujie Li, Multiple critical points of periodic functional and some applications, International Center for Theoretical Physics Tech. Rep. IC-86-191.
  • [5] J. Franks, Generalizations of the Poincaré-Birkhoff theorem, preprint. MR 951509 (89m:54052)
  • [6] C. C. Conley and E. Zehnder, The Birkhoff-Lewis fixed point theorem and a conjecture of V. I. Arnold, Invent. Math. 73 (1983), 33-49. MR 707347 (85e:58044)
  • [7] P. H. Rabinowitz, Minimax methods in critical point theory with applications to differential equations, CBMS Regional Conf. Ser. in Math., no. 65, Amer. Math. Soc., Providence, R. I., 1986. MR 845785 (87j:58024)
  • [8] J. T. Schwartz, Nonlinear functional analysis, Gordon & Breach, New York, 1969. MR 0433481 (55:6457)
  • [9] G. W. Whitehead, Elements of homotopy theory, Springer-Verlag, 1978. MR 516508 (80b:55001)
  • [10] P. Grisvard, Elliptic problems in nonsmooth domains, Pitman, 1985. MR 775683 (86m:35044)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 34C25, 35J60, 58E05, 58F22

Retrieve articles in all journals with MSC: 34C25, 35J60, 58E05, 58F22

Additional Information

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

American Mathematical Society