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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