Global structure of positive solutions for a fourth-order boundary value problem with singular data

Abstract

We are concerned with a problem described the deformation of a simply supported beam of the form $$ u^{(4)}(x)+c(x)u(x) + \sum^p\_{i=1}c\_i\delta(x-x\_i)u(x) = \lambda a(u(x)) + \lambda\sum^q\_{j=1}a\_j(u(x))\delta(x-y\_j), \quad x\in (0,1), $$ $$ u(0)=u(1)=u''(0)=u''(1)=0, $$ where $\lambda$ is a positive parameter, $c\in C(\[0, 1],\mathbb{R})$, $c\_i \in \mathbb{R}$, $a, a\_j\in C(\[0,\infty),\[0,\infty))$, $i = 1, 2, \ldots, p$, $j= 1, 2, \ldots, q$, $p, q \in \mathbb{N}$. The Dirac delta impulses $\delta = \delta(x)$ are applied at given points $0 < x\_1 < x\_2 <\cdots < x\_p < 1$ and $0 < y\_1 < y\_2 < \cdots < y\_q < 1$. We investigate the global structure of positive solutions by the global bifurcation techniques.