Existence results of positive solutions for Kirchhoff type biharmonic equation via bifurcation methods*

Authors: JINXIANG WANG, DABIN WANG

Abstract: This paper is concerned with the existence of positive solutions for the fourth order Kirchhoff type problem $$ \left\{\begin{array}{ll} \Delta^{2}u-(a+b\int_\Omega|\nabla u|^2dx)\triangle u=\lambda f(u(x)),\ \ \text{in}\ \Omega,\\ u=\triangle u=0,\ \ \text{on}\ \partial\Omega,\\ \end{array} \right. $$ where $\Omega\subset \mathbb{R}^{N}$($N\geq 1$) is a bounded domain with smooth boundary $\partial \Omega$, $a>0, b\geq 0$ 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 elliptic equations 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 positive solution for all $\lambda>0$ and no positive solution for $\lambda<0$ by using bifurcation method.

Keywords: Kirchhoff type biharmonic equation, global bifurcation, positive solution

Full Text: PDF