Abstract
The paper investigates the structure and properties of the set S of all positive solutions to the singular Dirichlet boundary value problem u″(t) + au′(t)/t − au(t)/t 2 = f(t, u(t),u′(t)), u(0) = 0, u(T) = 0. Here a ∈ (−∞,−1) and f satisfies the local Carathéodory conditions on [0,T]×D, where D = [0,∞)×ℝ. It is shown that S c = {u ∈ S: u′(T) = −c} is nonempty and compact for each c ≥ 0 and S = ∪ c≥0 S c . The uniqueness of the problem is discussed. Having a special case of the problem, we introduce an ordering in S showing that the difference of any two solutions in S c ,c≥ 0, keeps its sign on [0,T]. An application to the equation v″(t) + kv′(t)/t = ψ(t)+g(t, v(t)), k ∈ (1,∞), is given.
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Rachůnková, I., Staněk, S. Properties of the set of positive solutions to Dirichlet boundary value problems with time singularities. centr.eur.j.math. 11, 112–132 (2013). https://doi.org/10.2478/s11533-012-0047-1
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DOI: https://doi.org/10.2478/s11533-012-0047-1