Nodal solutions for a nonlocal fourth order equation of Kirchhoff type

We study the bifurcation behavior of nodal solutions for the Kirchhoff type beam equation $\left(\begin{array}{cc}& u^{\prime \prime \prime \prime }-M\left({\int }_0^1{\left|u^{\prime }\right|}^{2}dx\right)u^{\prime \prime }=\lambda f(x,u),x\in (0,1),\\ & u\left(0\right)=u\left(1\right)=u^{\prime \prime }\left(0\right)=u^{\prime \prime }\left(1\right)=0,\end{array}\right)$ where $\lambda \in R$ is a parameter, $M:[0,\infty )\to [0,\infty )$ and $f:[0,1]\times R\to R$ are smooth functions. We obtain the existence of nodal solutions under some suitable conditions. The proof of our main result is based upon bifurcation techniques.