Abstract

In this paper, we investigate the global structure of positive solutions of

$$\begin{cases} u''''(x)=\lambda h(x)f(u(x)), & 0<x<1, \\ u(0)=u(1)=u'(0)=u'(1)=0,& \end{cases}$$

where \(\lambda > 0\) is a parameter, \(h\in C[0,1]\), \(f\in C[0,\infty)\) and \(f(s)>0\) for \(s>0\). We show that the problem has three positive solutions suggesting suitable conditions on the nonlinearity. Furthermore, we also establish the existence of infinitely many positive solutions. The proof is based on the bifurcation method.

中文翻译：

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具有夹梁边界条件的四阶问题正解的全局结构

摘要

在本文中，我们研究了正解的全局结构

$$\begin{cases} u''''(x)=\lambda h(x)f(u(x)), & 0<x<1, \\ u(0)=u(1)=u '(0)=u'(1)=0,& \end{cases}$$

其中\(\lambda > 0\)是一个参数，\(h\in C[0,1]\)，\(f\in C[0,\infty)\)和\(f(s)>0 \)为\(s>0\)。我们表明该问题具有三个正解，表明非线性的合适条件。此外，我们还建立了无限多个正解的存在性。证明是基于分叉法。