## Journal of Applied Mathematics

### 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 $Lu=p(t)f(t,u(t),{u}^{\prime \prime }(t))-g(t,u(t),{u}^{\prime \prime }(t)),0, with ${\alpha }_{i},{\beta }_{i},{\gamma }_{i},{\delta }_{i}\ge 0$ and ${\beta }_{i}{\gamma }_{i}+{\alpha }_{i}{\gamma }_{i}+{\alpha }_{i}{\delta }_{i}>0,\mathrm{ }\mathrm{ }i=1,2$, where $L$ denotes the linear operator $Lu:=({ru}^{\prime \prime \prime })\text{'}-q{u}^{\prime \prime },r\in {C}^{1}([0,1],(0,+\infty ))$, and $q\in C([0,1],[0,+\infty ))$. This equation is viewed as a perturbation of the fourth-order Sturm-Liouville problem, where the perturbed term $g:(0,1){\times}[0,+\infty ){\times}(-\infty ,+\infty )\to (-\infty ,+\infty )$ 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.

#### Article information

Source
J. Appl. Math., Volume 2012 (2012), Article ID 160891, 23 pages.

Dates
First available in Project Euclid: 14 December 2012

https://projecteuclid.org/euclid.jam/1355495024

Digital Object Identifier
doi:10.1155/2012/160891

Mathematical Reviews number (MathSciNet)
MR2874980

Zentralblatt MATH identifier
1241.34026

#### Citation

Han, Yuefeng; Zhang, Xinguang; Liu, Lishan; Wu, Yonghong. Multiple Positive Solutions of Singular Nonlinear Sturm-Liouville Problems with Carathéodory Perturbed Term. J. Appl. Math. 2012 (2012), Article ID 160891, 23 pages. doi:10.1155/2012/160891. https://projecteuclid.org/euclid.jam/1355495024