Global bifurcation curves of nonlocal elliptic equations with oscillatory nonlinear term

TL;DR

This paper analyzes the bifurcation curves of a one-dimensional nonlocal Kirchhoff-type elliptic equation with oscillatory nonlinearities, deriving asymptotic formulas and revealing oscillatory behavior in the bifurcation parameter as the solution norm varies.

Contribution

It provides the first precise asymptotic descriptions of bifurcation curves for such nonlocal equations with oscillatory nonlinearities, including oscillatory behavior of the second term.

Findings

Asymptotic formulas for bifurcation curves as solution norm approaches zero and infinity.

Identification of oscillatory behavior in the bifurcation parameter at large solution norms.

Characterization of the second term in the bifurcation formula as oscillatory.

Abstract

We study the one-dimensional nonlocal elliptic equation of Kirchhoff type with oscillatory nonlinear term. We establish the precise asymptotic formulas for the bifurcation curves $\lambda(\alpha)$ as $\alpha \to \infty$ and $\alpha \to 0$, where $\alpha:= \Vert u_\lambda\Vert_\infty$ and $u_\lambda$ is the solution associated with $\lambda$. We show that the second term of $\lambda(\alpha)$ is oscillatory as $\alpha \to \infty$.