Asymptotic Behavior of Bifurcation Curve for Sine-Gordon-Type Differential Equation

Abstract EN

We consider the nonlinear eigenvalue problems for the equation −u″(t)+sin u(t)=λu(t), u(t)>0, t∈I=:(0,1), u(0)=u(1)=0, where λ>0 is a parameter.

It is known that for a given ξ>0, there exists a unique solution pair (uξ,λ(ξ))∈C2(I¯)×ℝ+ with ∥uξ∥∞=ξ.

We establish the precise asymptotic formulas for bifurcation curve λ(ξ) as ξ→∞ and ξ→0 to see how the oscillation property of sin u has effect on the behavior of λ(ξ).

We also establish the precise asymptotic formula for bifurcation curve λ(α) (α=∥uλ∥2) to show the difference between λ(ξ) and λ(α).