Flow-invariant sets and critical point theory

Jingxian Sun; Shouchuan Hu

doi:10.3934/dcds.2003.9.483

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2003, Volume 9, Issue 2: 483-496. Doi: 10.3934/dcds.2003.9.483

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Flow-invariant sets and critical point theory

Jingxian Sun^1, and
Shouchuan Hu^2,

1.
Department of Mathematics, Xuzhou Normal University, Xuzhou 221116
2.
Department of Mathematics, Southwest Missouri State University, Springfield, MO 65804

Received: December 31, 2000

Revised: August 31, 2002

Published: February 2003

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Abstract

In this paper, we study the relationship between flow-invariant sets for an vector field $-f'(x)$ in a Banach space, and the critical points of the functional $f(x)$. The Mountain-Pass Lemma, for functionals defined on a Banach space, is generalized to a more general setting where the domain of the functional $f$ can be any flow-invariant set for $-f'(x)$. Furthermore, the intuitive approach taken in the proofs provides a new technique in proving multiple critical points.

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Mathematics Subject Classification: 35B38, 34G20, 35A15.

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