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Zbl 1194.58008
Bonanno, Gabriele; Marano, Salvatore A.
On the structure of the critical set of non-differentiable functions with a weak compactness condition.
(English)
[J] Appl. Anal. 89, No. 1, 1-10 (2010). ISSN 0003-6811; ISSN 1563-504X/e

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 {\it R. Livrea} and {\it S. A. Marano} [Commun. Pure Appl. Anal. 8, No. 3, 1019--1029 (2009; Zbl 1208.58014)]. \par 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: $f_\lambda:=\Phi-\lambda\Psi$, with $\lambda$ a real parameter, $\Phi$ and $-\Psi$ sequentially weakly lower semi-continuous.
[Marlène Frigon (Montréal)]
MSC 2000:
*58E05 Abstract critical point theory
49J35 Minimax problems (existence)
49J52 Nonsmooth analysis (other weak concepts of optimality)

Keywords: critical set; locally Lipschitz continuous functions; weak Palais-Smale condition

Citations: Zbl 1208.58014

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