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

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.

### 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
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### References:

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