# American Institute of Mathematical Sciences

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

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