Asymptotic Behavior and Global Structure of Oscillatory Bifurcation Diagrams

Abstract

This paper is concerned with the nonlinear eigenvalue problem $$\begin{aligned} -u''(t) = \lambda f(u(t)), u(t) > 0, t \in I := (-1,1), u(\pm 1) = 0. \end{aligned}$$Here, $$f(u) = u^{2n+1} + \frac{\sin (u^2)}{u}$$ ($$n = 0,1,2, \ldots $$) and $$\lambda > 0$$ is a bifurcation parameter. Since $$f(u) > 0$$ for $$u > 0$$, $$\lambda $$ is a continuous function of the maximum norm $$\alpha = \Vert u_\lambda \Vert _\infty $$ of the solution $$u_\lambda $$ associated with $$\lambda $$, and is expressed as $$\lambda = \lambda (\alpha )$$. In this paper, by the argument of the stationary phase method, we establish the precise asymptotic formulas for $$\lambda (\alpha )$$ as $$\alpha \rightarrow \infty $$, which seem to be new, and $$\alpha \rightarrow 0$$ for the better understanding the global structure of $$\lambda (\alpha )$$.