Existence and uniqueness of positive solutions for Kirchhoff type beam equations

This paper is concerned with the existence and uniqueness of positive solution for the fourth order Kirchhoff type problem $\left\{\begin{array}{ll} u''''(x)-(a+b\int_0^1(u'(x))^2dx)u''(x)=λf(u(x)),\ \ \ \ x\in(0,1),\\ u(0)=u(1)=u''(0)=u''(1)=0,\\ \end{array} \right.$ where $a>0, b\geq 0$ are constants, $λ\in \mathbb{R}$ is a parameter. For the case $f(u)\equiv u$, we use an argument based on the linear eigenvalue problems of fourth order equations and their properties to show that there exists a unique positive solution for all $λ>λ_{1,a}$, here $λ_{1,a}$ is the first eigenvalue of the above problem with $b=0$; For the case $f$ is sublinear, we prove that there exists a unique positive solution for all $λ>0$ and no positive solution for $λ<0$ by using bifurcation method.

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