ABSTRACT
F = |f |nf =⇒ F ′′′ = −∣∣F ∣∣− nn+1F. (28) We describe oscillatory solutions of changing sign of (28), with zeros concentrating at the given interface point y = 0+. Oscillatory properties of solutions are a common feature of higher-order degenerate ODEs, which was first clearly detected in the pioneering papers by Bernis-McLeod [31, 33]. More clearly than before, we mention that, for such third-order degenerate ODEs, these authors proved that the solutions are compactly supported and are oscillatory near interfaces; see further, more detailed comments below. However, the character (a type) of such “nonlinear” oscillations was not detected before. Thus, similar to the approach to degenerate ODEs in Chapter 1, we are
going to give a sharp description of the behavior of the solutions close to interfaces. By the scaling invariance of (28), we look for solutions of the form
F (y) = yμϕ(s), s = ln y, where μ = 3(n+1)n > 3 for n > 0, (29)
where ϕ(s) is an oscillatory component. Substituting (29) into (28) yields
P3(ϕ) = −|ϕ|− nn+1ϕ. (30) Linear differential operators Pk are given by the recursion (as in Chapter 1)
Pk+1(ϕ) = P ′ k(ϕ) + (μ− k)Pk(ϕ), k ≥ 0; P0(ϕ) = ϕ, so that
P1(ϕ) = ϕ ′ + μϕ, P2(ϕ) = ϕ′′ + (2μ− 1)ϕ′ + μ(μ− 1)ϕ,
P3(ϕ) = ϕ ′′′ + 3(μ− 1)ϕ′′ + (3μ2 − 6μ+ 2)ϕ′ + μ(μ− 1)(μ− 2)ϕ, etc.
