论文标题
Kirchhoff类型梁方程的积极解决方案的存在和独特性
Existence and uniqueness of positive solutions for Kirchhoff type beam equations
论文作者
论文摘要
本文涉及第四阶Kirchhoff类型问题的积极解决方案的存在和独特性，$$ \ left \ weet {\ begin {array} {ll} {ll} u'''''（x） - （a+b \ int_0^1（a+b \ int_0^1（u'（x）） x \ in（0,1），\\ u（0）= u（1）= u'''（0）= u''（1）= 0，\\ \\ end {array} \ right。 $$其中$ a> 0，b \ geq 0 $是常数，$λ\ in \ mathbb {r} $是一个参数。对于$ f（u）\ equiv u $，我们使用基于第四阶方程的线性特征值问题及其属性的参数，以表明所有$λ>λ_{1，a} $，a} $，这里是$λ_{1，a} $的独特积极解决方案。对于$ f $的情况是sublinear，我们证明，使用分叉方法，对于所有$λ> 0 $，对于所有$λ> 0 $，对于$λ<0 $，没有正面解决方案。
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.
