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

Abstract

We consider the nonlinear eigenvalue problems for the equation $-u^{″}(t)+\text{sin\hspace{0.17em}}u(t)=\lambda u(t)$, $u(t)>0$, $t\in I=:(0,1)$, $u(0)=u(1)=0$, where $\lambda >0$ is a parameter. It is known that for a given $\xi >0$, there exists a unique solution pair $(u_{\xi },\lambda (\xi ))\in C^2(\overline{I})\times {ℝ}_{+}$ with ${\parallel u_{\xi }\parallel }_{\infty }=\xi$. We establish the precise asymptotic formulas for bifurcation curve $\lambda (\xi )$ as $\xi \to \infty$ and $\xi \to 0$ to see how the oscillation property of $\text{sin\hspace{0.17em}}u$ has effect on the behavior of $\lambda (\xi )$. We also establish the precise asymptotic formula for bifurcation curve $\lambda (\alpha )$ $(\alpha ={\parallel u_{\lambda }\parallel }_2)$ to show the difference between $\lambda (\xi )$ and $\lambda (\alpha )$.