Eigenvalues, global bifurcation and positive solutions for a class of nonlocal elliptic equations

Abstract

In this paper, we shall study global bifurcation phenomenon for the following Kirchhoff type problem: \begin{equation*} \begin{cases} \displaystyle -\bigg(a+b\int_\Omega \vert \nabla u\vert^2dx\bigg)\Delta u=\lambda u+h(x,u,\lambda) &\text{in } \Omega,\\ u=0 &\text{on }\Omega. \end{cases} \end{equation*} Under some natural hypotheses on $h$, we show that $(a\lambda_1,0)$ is a bifurcation point of the above problem. As an application 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 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 provide a positive answer to an open problem involving the case $a=0$.