In this paper, using a recent result by J. Saint Raymond ([6]), we improve the three critical points theorem established in [5]. Very recently,in [5], the following three critical points result has been established:Theorem A. Let X be a separable and reflexive real Banach space; F X X 3 R a continuously GaÃteaux differentiable and convex functional whose GaÃteaux derivative admits a continuous inverse on X à ; Y X X 3 R a continuously GaÃteaux differentiable functional whose GaÃteaux derivative is compact; I 7 R an interval; m 0 P I. Assume that lim kxk3 I Fx l À m 0 Yx I for all l P I, and that there exists a continuous concave function h X I 3 R such that sup lPI inf xPX Fx l À m 0 Yx hl`inf xPX sup lPI Fx l À m 0 Yx hl X Then, there exists l à P I nfm 0 g such that the equation F H x l à À m 0 Y H x 0 has at least three solutions in X.The aim of the present paper is to improve Theorem A in a twofold way. Namely, we wish to show that convexity of F can be replaced by sequential weak lower semicontinuity and, at the same time, that the conclusion even holds for each l à in an open sub-interval of I.So, we wish to prove the following Theorem 1. Let X be a separable and reflexive real Banach space; F X X 3 R a continuously GaÃteaux differentiable and sequentially weakly lower semicontinuous functional whose GaÃteaux derivative admits a continuous inverse on X à ; Y X X 3 R a continuously GaÃteaux differentiable functional whose GaÃteaux derivative is compact; I 7 R an interval. Assume that lim kxk3 I Fx lYx I Mathematics Subject Classification (1991): 35J20, 49J35, 58E05.
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