Asymptotic Behavior of the Bifurcation Diagrams for Semilinear Problems with Application to Inverse Bifurcation Problems

Abstract

We consider the nonlinear eigenvalue problem $u^{\mathrm{″}}(t)+\lambda f(u(t))=0$, $u(t)>0$, $t\in I=:(-\mathrm{1,1})$, $u(1)=u(-1)=0$, where $f(u)$ is a cubic-like nonlinear term and $\lambda >0$ is a parameter. It is known by Korman et al. (2005) that, under the suitable conditions on $f(u)$, there exist exactly three bifurcation branches $\lambda ={\lambda }_j(\xi )$ ($j=\mathrm{1,2},3$), and these curves are parameterized by the maximum norm $\xi$ of the solution $u_{\lambda }$ corresponding to $\lambda$. In this paper, we establish the precise global structures for ${\lambda }_j(\xi )$ ($j=\mathrm{1,2},3$), which can be applied to the inverse bifurcation problems. The precise local structures for ${\lambda }_j(\xi )$ ($j=\mathrm{1,2},3$) are also discussed. Furthermore, we establish the asymptotic shape of the spike layer solution $u_2(\lambda ,t)$, which corresponds to $\lambda ={\lambda }_2(\xi )$, as $\lambda \to \mathrm{\infty }$.