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Zbl 1136.34021
Xu, Fuyi; Su, Hua; Zhang, Xiaoyan
Positive solutions of fourth-order nonlinear singular boundary value problems. (English)
[J] Nonlinear Anal., Theory Methods Appl. 68, No. 5, A, 1284-1297 (2008). ISSN 0362-546X

Summary: We consider the existence of positive solutions for the following fourth-order singular Sturm-Liouville boundary value problem: $$\cases \frac{1}{p(t)}\,(p(t)u'''(t))'-g(t)F(t,u)=0,\quad 0<t<1,\\ \alpha_1u(0)-\beta_1u'(0)=0,\quad \gamma_1 u(1)+\delta_1u'(1)=0,\\ \alpha_2u''(0)-\beta_2\lim_{t\to 0+}p(t)u'''(t)=0,\\ \gamma_2u''(1)+\delta_2\lim_{t\to 1-}p(t)u'''(t)=0\endcases$$ where $g$, $p$ may be singular at $t=0$ and/or 1. Moreover $F(t,x)$ may also have singularity at $x=0.$ Existence and multiplicity theorems of positive solutions for the fourth-order singular Sturm-Liouville boundary value problem are obtained by using the first eigenvalue of the corresponding linear problems. Our results significantly extend and improve many known results including singular and nonsingular cases.

MSC 2000:: *34B16 Singular nonlinear boundary value problems
34B18 Positive solutions of nonlinear boundary value problems

Keywords: fourth-order singular differential equation; cone; positive solutions; fixed point theory

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