Global and local structures of oscillatory bifurcation curves with application to inverse bifurcation problem

Abstract

We consider the bifurcation problem $$ -u''(t) = \lambda (u(t) + g(u(t))), \quad u(t) > 0, \quad t \in I := (-1,1), \quad u(\pm 1) = 0, $$ where $g(u) = g_1(u) := \sin \sqrt{u}$ and $g_2(u) := \sin u^2 (= \sin (u^2))$, 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 $\lambda = \lambda(g,\alpha)$. When $g(u) = g_1(u)$, this problem has been proposed in Cheng [On an open problem of Ambrosetti, Brezis and Cerami, Differential Integral Equations 15 (2002), 1025-1044] as an example which has arbitrary many solutions near $\lambda = \pi^2/4$. We show that the bifurcation diagram of $\lambda(g_1,\alpha)$ intersects the line $\lambda = \pi^2/4$ infinitely many times by establishing the precise asymptotic formula for $\lambda(g_1,\alpha)$ as $\alpha \to \infty$. We also establish the precise asymptotic formulas for $\lambda(g_i,\alpha)$ ($i = 1,2$) as $\alpha \to \infty$ and $\alpha \to 0$. We apply these results to the new concept of inverse bifurcation problems.