Unexpected Solutions of the Nehari Problem

Abstract

The Nehari characteristic numbersλn(a,b)are the minimal values of an integral functional associated with a boundary value problem (BVP) for nonlinear ordinary differential equation. In case of multiple solutions of the BVP, the problem of identifying of minimizers arises. It was observed earlier that for nonoscillatory (positive) solutions of BVP those with asymmetric shape can provide the minimal value to a functional. At the same time, an even solution with regular shape is not a minimizer. We show by constructing the example that the same phenomenon can be observed in the Nehari problem for the fifth characteristic numberλn(a,b)which is associated with oscillatory solutions of BVP (namely, with those having exactly four zeros in(a,b)).