Multiple Positive Solutions of Singular Nonlinear Sturm-Liouville Problems with Carathéodory Perturbed Term

Abstract

By employing a well-known fixed point theorem, we establish the existence of multiple positive solutions for the following fourth-order singular differential equation , with ${\alpha }_i,{\beta }_i,{\gamma }_i,{\delta }_i\ge 0$ and ${\beta }_i{\gamma }_i+{\alpha }_i{\gamma }_i+{\alpha }_i{\delta }_i>0,i=\mathrm{1,2}$, where $L$ denotes the linear operator $Lu:=\left({ru}^{\prime \prime \prime }\right)\text{'}-q{u}^{\prime \prime },r\in C^1\left(\right[\mathrm{0,1}],(0,+\infty \left)\right)$, and $q\in C\left(\right[\mathrm{0,1}],[0,+\infty \left)\right)$. This equation is viewed as a perturbation of the fourth-order Sturm-Liouville problem, where the perturbed term $g:\left(\mathrm{0,1}\right)\{\backslash times\}[0,+\infty )\{\backslash times\}(-\infty ,+\infty )\to (-\infty ,+\infty )$ only satisfies the global Carathéodory conditions, which implies that the perturbed effect of $g$ on $f$ is quite large so that the nonlinearity can tend to negative infinity at some singular points.