Existence and uniqueness of positive solutions for Kirchhoff type beam equations

TL;DR

This paper investigates the existence and uniqueness of positive solutions for a fourth-order Kirchhoff type beam equation, establishing conditions based on the parameter and the form of the nonlinearity.

Contribution

It provides new results on positive solutions for Kirchhoff equations, including cases with linear and sublinear nonlinearities, using eigenvalue and bifurcation methods.

Findings

Unique positive solution for all λ > λ₁,a when f(u) ≡ u

Existence of a unique positive solution for all λ > 0 with sublinear f

No positive solutions for λ < 0 with sublinear f

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.