×

zbMATH — the first resource for mathematics

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
PDF BibTeX XML Cite
Full Text: DOI