Existence and uniqueness of positive solutions for Kirchhoff type beam equations

Abstract

This paper is concerned with the existence and uniqueness of positive solution for the fourth order Kirchhoff type problem $$\left\{\begin{array}{ll} u''''(x)-(a+b\int_0^1(u'(x))^2dx)u''(x)=\lambda f(u(x)),\ \ \ \ x\in(0,1),\\ u(0)=u(1)=u''(0)=u''(1)=0,\\ \end{array} \right. $$ where $a>0, b\geq 0$ are constants, $\lambda\in \mathbb{R}$ 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 $\lambda>\lambda_{1,a}$, here $\lambda_{1,a}$ 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 $\lambda>0$ and no positive solution for $\lambda<0$ by using bifurcation method.