On some generalizations of the Palais-Smale condition

In this paper we study the weak Palais-Smale condition introduced in [C. K. Zhong-A generalization of Ekeland's variational principle and application to the study of the relation between the weak PS condition and coercivity, Nonlinear Anal., Theory Methods Appl. 29 (1997) 1421-1431], the generalized Cerami condition of [A. R. El Amrouss-Critical point theorems and applications to differential equations, Acta Math. Sin., Engl. Ser. 21 (2005) 129-142] and the smooth version of the compactness condition used in [A. Kristaly, V. V. Motreanu, Cs. Varga-A minimax principle with a general Palais-Smale condition, Commun. Appl. Anal. 9 (2005) 285-297]. Then a generalization of a previous results of the author is given.