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On the structure of the critical set of non-differentiable functions with a weak compactness condition. (English) Zbl 1194.58008

The authors study the nature of critical points for locally Lipschitz continuous functionals satisfying a non-smooth Cerami condition. Critical points are obtained via min-max results under Ghoussoub’s type linking conditions. Their results rely on Theorem 3.1 of R. Livrea and S. A. Marano [Commun. Pure Appl. Anal. 8, No. 3, 1019–1029 (2009; Zbl 1208.58014)].

Assuming the existence of two local minimums (and other suitable hypotheses), the existence of a third critical point which is not a local minimum is established. This last result is applied to obtain the existence of three critical points of functionals of the form: f λ :=Φ-λΨ, with λ a real parameter, Φ and -Ψ sequentially weakly lower semi-continuous.

58E05Abstract critical point theory
49J35Minimax problems (existence)
49J52Nonsmooth analysis (other weak concepts of optimality)