# Reversed S-shaped connected component for a fourth-order boundary value problem

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, 2017:113

First Online: 18 April 2017Received: 17 October 2016Accepted: 04 April 2017DOI: 10.1186-s13662-017-1167-5

Abstract

In this paper, we investigate the existence of a reversed S-shaped component in the positive solutions set of the fourth-order boundary value problem $$\textstyle\begin{cases} u-x=\lambda hxfux,\quad x\in0,1,\\ u0=u1=u-0=u-1=0, \end{cases}$$ where \\lambda>0\ is a parameter, \h\in C0,1\ and \f\in C0,\infty \, \f0=0\, \fs>0\ for all \s>0\. By figuring the shape of unbounded continua of solutions, we show the existence and multiplicity of positive solutions with respect to parameter λ, and especially, we obtain the existence of three distinct positive solutions for λ being in a certain interval.Keywordsboundary value problem positive solutions principal eigenvalue bifurcation MSC34B10 34B18

Autor: Jinxiang Wang - Ruyun Ma