Asymptotic behavior of bifurcation curve for Sine-Gordon-type differential equation (Q1938293)

Summary: We consider the nonlinear eigenvalue problem \[ \begin{aligned} -u''(t) + \sin \,&u(t) = \lambda u(t),\;u(t) > 0,\quad t \in I =: (0, 1),\\ &u(0) = u(1) = 0,\end{aligned} \] where \(\lambda > 0\) is a parameter. It is known that, for a given \(\xi > 0\), there exists a unique solution pair \((u_\xi, \lambda(\xi)) \in C^2(\bar{I}) \times \mathbb R_+\) with \(\|u_\xi\|_\infty = \xi\). We establish precise asymptotic formulas for the bifurcation curve \(\lambda(\xi)\) as \(\xi \rightarrow \infty\) and \(\xi \rightarrow 0\) in order to see how the oscillation property of \(\sin u\) effects the behavior of \(\lambda(\xi)\). We also establish the precise asymptotic formula for the bifurcation curve \(\lambda(\alpha)(\alpha = \|u_\lambda\|_2)\) in order to show the difference between \(\lambda(\xi)\) and \(\lambda(\alpha)\).