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On the existence of positive solutions of nonlinear differential equations of high order. (English) Zbl 0990.34025

Here, the authors prove, that the infinitely many solutions result by Zhao is applicable to the following nonlinear differential equation \[ L^2= L(Lu)= -f(.,u)\quad\text{in }(0,\omega), \] with \(\omega\in (0,\infty]\), \(f\) a measurable function on \((0,\omega)\times (0,\infty)\) dominated by a regular function, and \(L\) the differential operator of second order defined on \((0,\omega)\) by \(Lu={1\over A} (Au')'\), where \(A\) is a continuous function on \([0,\omega)\), infinitely differentiable and positive on \((0,\omega)\).

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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References:

[1] Dalmasso, R., On singular nonlinear elliptic problems of second and fourth orders, Bull. Sci. Math., 116, 2, 95-110 (1992) · Zbl 0809.35024
[2] H. Maâgli, S. Masmoudi, Sur les solutions d’un opérateur différentiel singulier semi-linéaire, Potential Analysis, to appear.; H. Maâgli, S. Masmoudi, Sur les solutions d’un opérateur différentiel singulier semi-linéaire, Potential Analysis, to appear.
[3] Zhao, Z., Positive solutions of nonlinear second order ordinary differential equations, Proc. Amer. Math. Soc., 121, 465-469 (1994) · Zbl 0802.34026
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