Asymptotic formulas for oscillatory bifurcation diagrams of semilinear ordinary differential equations
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We study the nonlinear eigenvalue problem
-u″(t) = λ(u(t)p + g(u(t))), u(t) > 0, t ∈ (-1, 1), u(±1) = 0,
where g(u) = h(u) sin(ur), p, r are given constants satisfying p ≥ 0, 0 < r ≤ 1 and λ > 0 is a parameter. It is known that under suitable conditions on h, λ is parameterized by the maximum norm α = ∥uα∥∞ of the solution uλ associated with λ and λ = λ(α) is a continuous function for α > 0. When p = 1, h(u) ≡ 1 and r = 1/2, this equation has been introduced by Chen  as a model equation such that there exist infinitely many solutions near λ = π2/4. We prove that λ(α). It is found that the shapes of bifurcation curves depend on the condition p > 1 or p < 1. The main tools of the proof are time-map argument and stationary phase method.
CitationShibata, T. (2019). Asymptotic formulas for oscillatory bifurcation diagrams of semilinear ordinary differential equations. Electronic Journal of Differential Equations, 2019(62), pp. 1-11.
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