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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