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

Abstract

We consider the nonlinear eigenvalue problemu″(t)+λf(u(t))=0,  u(t)>0,  t∈I=:(-1,1),  u(1)=u(-1)=0, wheref(u)is a cubic-like nonlinear term andλ>0is a parameter. It is known by Korman et al. (2005) that, under the suitable conditions onf(u), there exist exactly three bifurcation branchesλ=λj(ξ)(j=1,2,3), and these curves are parameterized by the maximum normξof the solutionuλcorresponding toλ. In this paper, we establish the precise global structures forλj(ξ)(j=1,2,3), which can be applied to the inverse bifurcation problems. The precise local structures forλj(ξ)(j=1,2,3) are also discussed. Furthermore, we establish the asymptotic shape of the spike layer solutionu2(λ,t), which corresponds toλ=λ2(ξ), asλ→∞.