Characteristic Numbers of Non‐autonomous Emden‐fowler Type Equations

Abstract

We consider the Emden‐Fowler equation x” = ‐q(t)|x|2εx, ε > 0, in the interval [a,b]. The coefficient q(t) is a positive valued continuous function. The Nehari characteristic number An associated with the Emden‐Fowler equation coincides with a minimal value of the functional [] over all solutions of the boundary value problem x” = ‐q(t)|x|2εx, x(a) = x(b) = 0, x(t) has exactly (n ‐ 1) zeros in (a, b). The respective solution is called the Nehari solution. We construct an example which shows that the Nehari extremal problem may have more than one solution.

Unexpected Solutions of the Nehari Problem

“…The respective values of the Nehari functional ( ) (5) were calculated and the result is the same: the even solution supplies ( even ) = 505549, the two solutions of asymmetric shape supply the value ( asym ) = 332861. Therefore, we confirm the phenomenon observed in [5,6] also for oscillatory (with exactly two zeros in (−1, 1)) solutions.…”

“…These minimizers are positive solutions of the problem (4) ( = 1). 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.…”

On boundary value problems with prescribed number of zeroes of solutions

“…The Nehari theory for Emden-Fowler type equations (1.1) provides a way to detect the existence of solutions with prescribed number of zeros [4][5][6]10,14]. For instance, in the case of the Dirichlet boundary value problem and˛is chosen in a specific way, the Nehari theory gives three positive solutions of (1.2).…”

“…For instance, in the case of the Dirichlet boundary value problem and˛is chosen in a specific way, the Nehari theory gives three positive solutions of (1.2). In [4], the authors construct three positive solutions of problem (1.2) for q.t/ D 2=.1 C h .t // 6 with an h > 0 and the "^-shaped" piecewise linear ,…”

“…The methods used to construct the solution usually strongly depend on the specific form of the equation. In particular, the approach of [4] is based on the explicit use of the lemniscate functions sl t and cl t , i. e., the respective solutions of the initial value problems u.0/ D 0, u 0 .0/ D 1 and u.0/ D 1, u 0 .0/ D 0 for the equation u 00 D 2u 3 . Motivated, in particular, by the work [4] on the above-mentioned problem of form (1.2), we show here how parametrisation techniques can be used to construct solutions of this kind of equations.…”