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A new proof based on Bishop-Phelps’ variational principle is given to a critical point theorem of Schechter for extrema in a ball of a Hilbert space. The same technique is used to obtain a similar result in annular domains. Comments on the involved boundary conditions and an application to a two-point boundary value problem are included. An alternative variational approach to the compression-expansion Krasnoselskii’s fixed point method is thus provided. In addition, estimations from below are obtained here for the first time, in terms of the energetic norm.

Mathematics Subject Classification (2010): 47J30, 58E05, 34B15.

Keywords: Critical point, extremum point, Palais-Smale condition, two-point boundary value problem.