Extension of the example by Moore-Nehari

Abstract

ABSTRACT. R. Moore and Z. Nehari developed the variational theory for superlinear boundary value problems of the form x = −p(t)|x| 2ε x, x(a) = 0 = x(b), where ε > 0 and p(t) is a positive continuous function. They constructed simple example of the equation considered in the interval [0, b] so that the problem had three positive solutions. We show that this example can be extended so that the respective BVP has infinitely many groups of solutions with a presribed number of zeros. The example was fairly simple: the coefficient q(t) was chosen as a step-wise functionTwo triples of symmetric solutions given on the intervals [0, 1] and [α, α + 1] can be connected by the straight line segments (solutions of x = 0) if the length of the middle interval [1, α] is chosen appropriately (Fig. 1, 2). This is the main idea of the example.

Wolfram Mathematica application to determination of the number of solutions for certain nonlinear boundary value problems

“…In our problem, we are dealing with three parameters a, b and δ and their influence on the number of solutions. There are multiple articles devoted to the study of differential equations, combined of several ones on disjoint subintervals of the main interval, for example, (Gritsans & Sadyrbaev, 2015), (Ellero & Zanolin, 2013), (Kirichuka and Sadyrbaev, 2018), (Kirichuka, 2016), (Moore & Nehari, 1959). In the paper (Kirichuka & Sadyrbaev, 2018a) an equation with cubic nonlinearity and step-wise potentials were studied together with the Dirichlet conditions.…”

Symmetry-breaking bifurcation for the Moore–Nehari differential equation

“…They proved that there exists a bifurcation point p ∈ (1, p λ ) with p λ := (N + 2 + 2λ)/(N − 2) such that a positive non-radial solution bifurcates from a unique positive radial solution and the bifurcation branch is unbounded in the Hölder space C 1,γ 0 (B). Gritsans and Sadyrbaev [5] investigated (1.1) when p = 3 and h(x, λ) = 0 for |x| < λ and h(x, λ) = 2 for λ ≤ |x| ≤ 1. They proved that for any λ ∈ (0, 1), (1.1) has infinitely many sign-changing solutions.…”

Existence of nodal solutions for the sublinear Moore-Nehari differential equation

“…Gritsans and Sadyrbaev [3] investigated (1.1) for p = 3 and proved that (1.1) has infinitely many sign-changing solutions. Kajikiya, Sim and Tanaka [10] studied the bifurcation problem of positive solutions for (1.1) with p > 1.…”