Oscillatory bifurcation for semilinear ordinary differential equations

Abstract

We consider the nonlinear eigenvalue problem \[u''(t) + \lambda f(u(t)) = 0, \quad u(t) > 0, \quad t \in I := (-1,1), \quad u(1) = u(-1) = 0, \] where $f(u) = u + (1/2)\sin^k u$ ($k \ge 2$) and $\lambda > 0$ is a bifurcation parameter. It is known that $\lambda$ is parameterized by the maximum norm $\alpha = \Vert u_\lambda\Vert_\infty$ of the solution $u_\lambda$ associated with $\lambda$ and is written as $\lambda = \lambda(k,\alpha)$. When we focus on the asymptotic behavior of $\lambda(k,\alpha)$ as $\alpha \to \infty$, it is natural to expect that $\lambda(k, \alpha) \to \pi^2/4$, and its convergence rate is common to $k$. Contrary to this expectation, we show that $\lambda(2n_1+1,\alpha)$ tends to $\pi^2/4$ faster than $\lambda(2n_2,\alpha)$ as $\alpha \to \infty$, where $n_1\ge 1,\ n_2 \ge 1$ are arbitrary given integers.

Keywords

• oscillatory bifurcation
• asymptotic behavior
• asymptotic length of bifurcation curves