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Nonlinear singular Sturm-Liouville problems. (English) Zbl 0944.34017
Singular Sturm-Liouville differential equations of the form $-(r(t)u')'+p(t)u=f(t,u),\quad t \in(a,b),$ associated with boundary conditions of Dirichlet type are considered.
In [C. de Coster, M. R. Grossinho and P. Habets, Appl. Anal. 59, No. 1-4, 241-256 (1995; Zbl 0847.34022)] conditions of the existence of at least two (positive) solutions in the case when $$\frac{1}{r} \in L^1(a,b)$$ and $$p(t)=0$$ are given.
It is proved, that this result can be adapted to the situation where $$p$$ is not necessarily $$0$$ and $$\frac{1}{r}$$ is not necessarily integrable.
Some results from H. G. Kaper, M. K. Kwong and A. Zettl [SIAM Rev. 17, 339-360 (1975)] are improved.

##### MSC:
 34B24 Sturm-Liouville theory 34B27 Green’s functions for ordinary differential equations 34-02 Research exposition (monographs, survey articles) pertaining to ordinary differential equations 34B15 Nonlinear boundary value problems for ordinary differential equations 34B18 Positive solutions to nonlinear boundary value problems for ordinary differential equations
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