Asymptotic behavior of the bifurcation diagrams for semilinear problems with application to inverse bifurcation problems (Q274818)

The author establishes the precise asymptotic formulas for bifurcation branches \(\lambda_{j}(\xi)\) \((j=1,2,3)\) of a nonlinear eigenvalue problem consisting of the equation NEWLINENEWLINE\[NEWLINE u''(t)+\lambda f(u(t))=0,\quad t\in I=(-1,1) NEWLINE\]NEWLINE NEWLINEwith the boundary conditions \(u(1)=u(-1)=0\).NEWLINENEWLINEHere, \(u(t)>0\) and \(f(u)\) is a cubic-like nonlinear term and \(\lambda >0\) is an eigenvalue parameter. The main goal of the paper is to clarify how the difference of the asymptotic shapes of solutions corresponding to these three curves gives effect to the asymptotic formulas for \(\lambda_{j}(\xi)\) \((j=1,2,3)\) and to establish the asymptotic shape of the spike layer solution \(u_{2}(\lambda,t)\), which corresponds to \(\lambda=\lambda_{2}(\xi)\) as \(\lambda \to \infty\).