Bifurcation diagrams of one-dimensional Kirchhoff type equations

TL;DR

This paper analyzes bifurcation diagrams and eigenvalues of one-dimensional Kirchhoff type equations, providing explicit solutions and complete bifurcation curve shapes, advancing understanding of nonlinear eigenvalue problems with variable parameters.

Contribution

It derives exact solutions and characterizes bifurcation curves for Kirchhoff type equations, and determines the first eigenvalue and eigenfunction using a simple time map method.

Findings

Explicit solution for the Kirchhoff equation established.

Complete shape of bifurcation curves determined.

First eigenvalue and eigenfunction identified.

Abstract

We study the one-dimensional Kirchhoff type equation $$ -(b + a\Vert u'\Vert^{2}) u''(x) = \lambda u(x)^p, x \in I:= (-1,1), \enskip u(x) > 0, \enskip x\in I, \enskip u(\pm 1) = 0, $$ where $\Vert u'\Vert = \left(\int_I u'(x)^2 dx\right)^{1/2}$, $a > 0, b > 0, p> 0$ are given constants and $\lambda > 0$ is a bifurcation parameter. We establish the exact solution $u_\lambda(x)$ and complete shape of the bifurcation curves $\lambda = \lambda(\xi)$, where $\xi:= \Vert u_\lambda\Vert_\infty$. We also study the nonlinear eigenvalue problem $$ -\Vert u'\Vert^{p-1} u''(x) = \mu u(x)^p, x \in I, \enskip u(x) > 0, x\in I, \enskip u(\pm 1) = 0, $$ where $p > 1$ is a given constant and $\mu > 0$ is an eigenvalue parameter. We establish the first eigenvalue and eigenfunction of this problem by using a simple time map method.