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Positive solutions of fourth-order nonlinear singular boundary value problems. (English) Zbl 1136.34021

Summary: We consider the existence of positive solutions for the following fourth-order singular Sturm-Liouville boundary value problem:
\[ \begin{cases} \frac{1}{p(t)}\,(p(t)u'''(t))'-g(t)F(t,u)=0,\quad 0<t<1,\\ \alpha_1u(0)-\beta_1u'(0)=0,\quad \gamma_1 u(1)+\delta_1u'(1)=0,\\ \alpha_2u''(0)-\beta_2\lim_{t\to 0+}p(t)u'''(t)=0,\\ \gamma_2u''(1)+\delta_2\lim_{t\to 1-}p(t)u'''(t)=0\end{cases} \]
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:

34B16 Singular nonlinear boundary value problems for ordinary differential equations
34B18 Positive solutions to nonlinear boundary value problems for ordinary differential equations
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