Eigenvalue, global bifurcation and positive solutions for a class of fully nonlinear problems

TL;DR

This paper investigates the global bifurcation and positive solutions for a class of nonlinear Kirchhoff problems, establishing bifurcation points, solution intervals, and eigenvalue properties under certain hypotheses.

Contribution

It introduces new results on bifurcation points and positive solutions for nonlinear Kirchhoff problems, including the case when a=0, and analyzes eigenvalue properties.

Findings

Identifies (aλ₁,0) as a bifurcation point.

Determines solution existence intervals for specific nonlinearities.

Provides properties of the first eigenvalue for a nonlocal problem.

Abstract

In this paper, we shall study global bifurcation phenomenon for the following Kirchhoff type problem \begin{equation} \left\{ \begin{array}{l} -\left(a+b\int_\Omega \vert \nabla u\vert^2\,dx\right)\Delta u=\lambda u+h(x,u,\lambda)\,\,\text{in}\,\, \Omega,\\ u=0~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\text{on}\,\,\Omega. \end{array} \right.\nonumber \end{equation} Under some natural hypotheses on $h$, we show that $\left(a\lambda_1,0\right)$ is a bifurcation point of the above problem. As applications of the above result, we shall determine the interval of $\lambda$, in which there exist positive solutions for the above problem with $h(x,u;\lambda)=\lambda f(x,u)-\lambda u$, where $f$ is asymptotically linear at zero and is asymptotically 3-linear at infinity. To study global structure of bifurcation branch, we also establish some properties of the first eigenvalue for a nonlocal eigenvalue problem. Moreover, we also provide a positive answer to an open problem involving the case of $a=0$.