Unexpected Solutions of the Nehari Problem

The Nehari characteristic numbers 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 number which is associated with oscillatory solutions of BVP (namely, with those having exactly four zeros in . 1. Introduction The variational theory of eigenvalues in Sturm-Liouville problems for linear ordinary differential equations provides variational interpretation of eigenvalues which emerge as minima of some quadratic functionals being considered with certain restrictions [1]. As to nonlinear boundary value problems for ordinary differential equations, the Nehari theory of characteristic values provides some analogue of the linear theory. The Nehari theory deals in particular with superlinear differential equations of the form The Nehari numbers , by definition, are minimal values of the functional over the set of all functions , which are continuous and piecewise continuously differentiable in ; there exist numbers such that and in any ; in any , and It was proved in [2] (see also [3]) that minimizers in the above variational problem are -solutions of the boundary value problem Putting (3) into (2) one gets where is an appropriate solution of the BVP (4). The BVP (4) may have multiple solutions but not all of them are minimizers. It appears that in order to detect it is sufficient to consider solutions of the boundary value problem (4). Z. Nehari posed the question is it true that there is only one minimizer associated with . It was shown implicitly in [4] that there may be multiple minimizers associated with the first characteristic number . These minimizers are positive solutions of the problem (4) ( ). Later in the work by the authors [5] the example was constructed showing three solutions of the BVP (4). They are depicted in Figure 1. Figure 1: Three solutions of the BVP from [ 5]. Two of these solutions are asymmetric and one is an even function. Surprisingly, two asymmetric solutions are the minimizers. The same phenomenon was observed later by Kajikiya [6] who studied “the Emden-Fowler equation whose coefficient is even in the