Global and local structures of oscillatory bifurcation curves

Abstract

We consider the nonlinear eigenvalue problem

where $g(u)=u^p\sin (u^q)$ ($0\le p<1$, $0<q\le 1$) and $\lambda >0$ is a bifurcation parameter. It is known that, in this case, $\lambda$ is parameterized by the maximum norm $\alpha =\Vert u_{\lambda }{\Vert }_{\infty }$ of the solution $u_{\lambda }$ associated with $\lambda$ and is written as $\lambda =\lambda (\alpha )$. We show that the bifurcation curve $\lambda (\alpha )$ intersects the line $\lambda ={\pi }^2/4$ infinitely many times by establishing the precise asymptotic formula for $\lambda (\alpha )$ as $\alpha \to \infty$ and $\alpha \to 0$. We find that, according to the relationship between $p$ and $q$, there exist three types of bifurcation curves.