Flow-invariant sets and critical point theory
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.
Connected set, Critical point, Forward saturated solution, Invariant set of flow
Sun, Jingxian, and Shouchuan Hu. "Flow-invariant sets and critical point theory." Discrete & Continuous Dynamical Systems-A 9, no. 2 (2003): 483.
Discrete and Continuous Dynamical Systems