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

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 $a=0$.