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Multiple positive solutions of semilinear differential equations with singularities. (English) Zbl 1032.34019
The author deals with the Sturm-Liouville boundary value problem associated to a second-order differential equation of the form ${u}^{\text{'}\text{'}}+g\left(t\right)f\left(u\right)=0$. Conditions are given for the existence of multiple positive solutions. The main features of the present paper are that it is assumed that $g$ is measurable (not necessarily integrable) and that no monotonicity assumption on $f$ is required. The proofs are performed by means of a fixed-point theorem in cones. Applications are provided to eigenvalue problems and to the search of radial solutions to elliptic equations. The results in this paper generalise various earlier contributions. See, among others, R. W. Leggett and L. R. Williams [Indiana Univ. Math. J. 28, 673-688 (1979; Zbl 0421.47033)], K. Lan and J. R. L. Webb [J. Differ. Equations 148, 407-421 (1998; Zbl 0909.34013)]; for related results obtained by the upper-lower solution method, we refer to A. K. Ben-Naoum and C. De Coster [Differ. Integral Equ. 10, 1093-1112 (1997; Zbl 0940.35086)] and M. Gaudenzi and P. Habets [Topol. Methods Nonlinear Anal. 14, 131-150 (1999; Zbl 0965.34011)].

##### MSC:
 34B18 Positive solutions of nonlinear boundary value problems for ODE 34B15 Nonlinear boundary value problems for ODE 35J65 Nonlinear boundary value problems for linear elliptic equations
##### Keywords:
multiple positive solutions; fixed-point index