Flow-invariant sets and critical point theory

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March 2003, 9(2): 483-496. doi: 10.3934/dcds.2003.9.483

Flow-invariant sets and critical point theory

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

Received January 2001 Revised September 2002 Published December 2002

Abstract
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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.

Keywords: connected set., Invariant set of flow, critical point, forward saturated solution.

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

Citation: Jingxian Sun, Shouchuan Hu. Flow-invariant sets and critical point theory. Discrete & Continuous Dynamical Systems - A, 2003, 9 (2) : 483-496. doi: 10.3934/dcds.2003.9.483

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