On an Ambrosetti–Prodi type problem for a class of fourth-order ODEs involving Dirac weights

The aim of this paper is to establish an Ambrosetti–Prodi type result involving Dirac weights

$$\begin{aligned} \left\{ \begin{array}{ll} u''''(x)+q(x)u(x)=(c (x)+\sum \limits _{i=1}^{p}c_{i}\delta (x-x_i))(g(u(x))+f(x)),~~~~&{}x\in (0,1),\\ \ u(0)=u(1)=u''(0)=u''(1)=0,\\ \end{array}\right. \end{aligned}$$

where \(\delta (x-x_i)\) is the canonical Dirac delta function at the point \(x_i\), \(i=1,2,\ldots ,p\), \(p\in \mathbb {N}\), \(0=x_0<x_1<\cdots<x_p<x_{p+1}=1\), \(q\in C([0,1],[0,+\infty ))\), \(f\in L^1([0,1],\mathbb {R})\), \(g\in C^{1}(\mathbb {R},\mathbb {R})\), \(c\in C([0,1],[0,+\infty ))\), \(c_i\in [0,+\infty )\). The main tools used are the sub-super-solution method and Leray–Schauder topological degree theory.