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Existence and nonexistence for a singular boundary value problem. (English) Zbl 0628.34025
Existence of positive solutions of the two-point boundary value problem \(y''=\lambda ^ 2f(x,y)\), \(y(0)=a>0\), \(y(1)=b\geq 0\) is studied, where \(\lambda ^ 2>0\) and \(\int _{0}f(x,y)dy<\infty\) but \(\lim _{y\to 0}f(x,y)=\infty.\) It is shown that positive solutions exist provided \(\lambda ^ 2\) is sufficiently small but that no positive solutions exist for \(\lambda ^ 2\) large.

MSC:
34B15 Nonlinear boundary value problems for ordinary differential equations
34C11 Growth and boundedness of solutions to ordinary differential equations
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