Wang, Jinxiang Existence and uniqueness of positive solutions for Kirchhoff type beam equations. (English) Zbl 1474.34184 Electron. J. Qual. Theory Differ. Equ. 2020, Paper No. 61, 14 p. (2020). Summary: This paper is concerned with the existence and uniqueness of positive solutions for the fourth order Kirchhoff type problem \[ u''''(x)-\Big(a+b\int_0^1(u'(x))^2dx\Big)u''(x)=\lambda f(u(x)),\; x\in(0,1),\] \[ u(0)=u(1)=u''(0)=u''(1)=0, \] 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. MSC: 34B18 Positive solutions to nonlinear boundary value problems for ordinary differential equations 34B10 Nonlocal and multipoint boundary value problems for ordinary differential equations 34B09 Boundary eigenvalue problems for ordinary differential equations 34C23 Bifurcation theory for ordinary differential equations 74K10 Rods (beams, columns, shafts, arches, rings, etc.) Keywords:fourth-order boundary value problem; Kirchhoff-type beam equation; global bifurcation; positive solution; uniqueness PDFBibTeX XMLCite \textit{J. Wang}, Electron. J. Qual. Theory Differ. Equ. 2020, Paper No. 61, 14 p. (2020; Zbl 1474.34184) Full Text: DOI arXiv