A critical points theorem and nonlinear differential problems. (English) Zbl 1087.58007

Both D. Averna and G. Bonanno [Topol. Methods Nonlinear Anal. 22, 93–103 (2003; Zbl 1048.58005)], and G. Bonanno [Nonlinear Anal., Theory Methods Appl. 54, 651–665 (2003; Zbl 1031.49006)] established a theorem on the existence of 3 critical points of a functional of the type \(\Phi - \lambda J\) for each \(\lambda\) in a suitable interval. In this paper, conditions are given ensuring that both theorems hold and hence the functional \(\Phi - \lambda J\) has 3 critical points for each \(\lambda\) in the union of those intervals. This result is then applied to the Dirichlet boundary value problem \(u'' + \lambda f(u) = 0\), \(u(0)=u(1)=0\).


58E05 Abstract critical point theory (Morse theory, Lyusternik-Shnirel’man theory, etc.) in infinite-dimensional spaces
34B15 Nonlinear boundary value problems for ordinary differential equations
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