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 equationLu=p(t)f(t,u(t),u′′(t))-g(t,u(t),u′′(t)),0<t<1,α1u(0)-β1u'(0)=0,γ1u(1)+δ1u'(1)=0,α2u′′(0)-β2u′′′(0)=0,γ2u′′(1)+δ2u′′′(1)=0, withαi,βi,γi,δi≥0andβiγi+αiγi+αiδi>0,    i=1,2, whereLdenotes the linear operatorLu:=(ru′′′)'-qu′′,r∈C1([0,1],(0,+∞)), andq∈C([0,1],[0,+∞)). This equation is viewed as a perturbation of the fourth-order Sturm-Liouville problem, where the perturbed termg:(0,1)×[0,+∞)×(-∞,+∞)→(-∞,+∞)only satisfies the global Carathéodory conditions, which implies that the perturbed effect ofgonfis quite large so that the nonlinearity can tend to negative infinity at some singular points.