On the relation between the weak Palais–Smale condition and coercivity given by Zhong

In this paper, we discuss Zhong’s result that the weak Palais–Smale condition implies coercivity under some assumption given in [C.-K. Zhong, A generalization of Ekeland’s variational principle and application to the study of the relation between the weak P.S. condition and coercivity, Nonlinear Anal. 29 (1997) 1421–1431]. We also give a simple proof of Zhong’s result. Further we generalize the result of Caklovic, Li and Willem [L. Caklovic, S.J. Li, M. Willem, A note on Palais–Smale condition and coercivity, Differential Integral Equations 3 (1990) 799–800].