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

Bonanno, Gabriele

J. Glob. Optim. 28, No. 3-4, 249-258 (2004).

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 \(u'' + \lambda f(u) = 0\), \(u(0)=u(1)=0\).

Reviewer: Marlène Frigon (Montréal)

MSC:

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

Keywords:

critical points; three solutions; two-point boundary value problem

Citations:

Zbl 1048.58005; Zbl 1031.49006