Spectrum, global bifurcation and nodal solutions to Kirchhoff-type equations

In this article, we consider a Dancer-type unilateral global bifurcation for the Kirchhoff-type problem $$\displaylines{ -\Big(a+b\int_0^1 | u'|^2\,dx\Big)u'' =\lambda u+h(x,u,\lambda)\quad\text{in } (0,1),\cr u(0)=u(1)=0. }$$ Under natural hypotheses on h, we show that $(a\lambda_k,0)$ is a bifurcation point of the above problem. As applications we determine the interval of $\lambda$, in which there exist nodal solutions for the Kirchhoff-type problem $$\displaylines{ -\Big(a+b\int_0^1 | u'|^2\,dx\Big) u'' =\lambda f(x,u)\quad\text{in } (0,1),\cr u(0)=u(1)=0, }$$ where f is asymptotically linear at zero and is asymptotically 3-linear at infinity. To do this, we also establish a complete characterization of the spectrum of a nonlocal eigenvalue problem.