A minimax inequality and its applications to ordinary differential equations. (English) Zbl 1009.49004

Bonanno, Gabriele

J. Math. Anal. Appl. 270, No. 1, 210-229 (2002).

The aim of this paper is twofold. First, the author establishes several abstract results related to the minimax inequality which is involved in the three critical points theorem of Ricceri. Equivalent formulations are shown, and characterization is argued for a special class of functionals. In the second part of the paper the author applies these results in the study of the nonlinear eigenvalue differential equation \(u''+\lambda f(u)=0\), subject to the Dirichlet conditions \(u(0)=u(1)=0\), where \(f:\mathbb{R}\rightarrow \mathbb{R}\) is a continuous function.

Reviewer: Vicenţiu D.Rădulescu (Craiova)

MSC:

49J35	Existence of solutions for minimax problems
34A40	Differential inequalities involving functions of a single real variable
58E05	Abstract critical point theory (Morse theory, Lyusternik-Shnirel’man theory, etc.) in infinite-dimensional spaces
47E05	General theory of ordinary differential operators

Keywords:

minimax inequality; critical point; multiplicity result

References:

[1]	Ricceri, B., On a three critical points theorem, Arch. math. (basel), 75, 220-226, (2000) · Zbl 0979.35040
[2]	Cordaro, G., On a minimax problem of ricceri, J. inequal. app., 6, 261-285, (2001) · Zbl 0986.49003
[3]	Avery, R.I.; Henderson, J., Three symmetric positive solutions for a second-order boundary value problem, Appl. math. lett., 13, 1-7, (2000) · Zbl 0961.34014
[4]	Bonanno, G., Existence of three solutions for a two point boundary value problem, Appl. math. lett., 13, 53-57, (2000) · Zbl 1009.34019
[5]	Henderson, J.; Thompson, H.B., Existence of multiple solutions for second order boundary value problems, J. differential equations, 166, 443-454, (2000) · Zbl 1013.34017
[6]	Korman, P.; Ouyang, T., Exact multiplicity results for a class of boundary value problems with cubic nonlinearities, J. math. anal. appl., 194, 328-341, (1995) · Zbl 0837.34033
[7]	Korman, P.; Ouyang, T., Exact multiplicity results for two classes of boundary value problem, Diff. integral equations, 6, 1507-1517, (1993) · Zbl 0780.34013
[8]	B. Ricceri, Existence of three solutions for a class of elliptic eigenvalue problems, Math. Comput. Modelling 32 1485-1494 · Zbl 0970.35089
[9]	Gelfand, I.M., Some problems in the theory of quasilinear equations, (), 295-381 · Zbl 0127.04901
[10]	Candito, P., Existence of three solutions for a nonautonomous two point boundary value problem, J. math. anal. appl., 252, 532-537, (2000) · Zbl 0980.34015
[11]	R. Livrea, Existence of three solutions for a quasilinear two point boundary value problem, Arch. Math. (Basel), to appear · Zbl 1015.34012