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On the exact number of solutions of a singular boundary-value problem. (English) Zbl 1240.34106
The paper investigates the Dirichlet problem \[ u''+f(u)=0,\;u(-R)=u(R)=0, \] where \(f(u)=-u^{-\gamma }+\beta \), \(\gamma \in (0,1)\), \(\beta >0\), \(R>0\). In particular, the exact number of positive solutions to the problem is determined. The authors derive a positive number \(R_0\) depending on \(\gamma \) and \(\beta \) and prove that, in the case \(1/2\leq \gamma <1\), there is no solution to the above problem for \(R<R_0\), and there is a unique solution for \(R\geq R_0\). In the case \(0<\gamma <1/2\), there exists \(R_{\text{min}}<R_0\) such that there is no solution to the above problem for \(R<R_{\text{min}}\), there is a unique solution for \(R=R_{\text{min}}\), there are exactly two solutions for \(R\in (R_{\text{min}},R_0]\), and there exists exactly one solution for \(R>R_0\). The proof is based on the shooting method and on properties of a time-map \(T\) which associates the first root of a solution \(u\) of the initial problem \[ u''+f(u)=0,\;u(0)=p,\;u'(0)=0 \] to its initial value, i.e., \(u(T(p))=0\). Hence, \(u\) is a positive solution to the above problem if and only if \(T(p)=R\). The properties of the time-map are proved for a general nonlinearity \(f\) that has a similar shape as \(f(u)=-u^{-\gamma }+\beta \).

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