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Upper and lower solutions method and a superlinear singular boundary value problem. (English) Zbl 1058.34021

The paper is devoted to the solvability of a singular Dirichlet boundary value problem of the form \[ u''= g(t,u,u'),\quad u(0)=u(1)=0. \tag{1} \] The nonlineearity \(g: (0,1)\times (0,\infty)\times \mathbb{R} \to \mathbb{R}\) is continuous, can change its sign and can have time singularities at \(t=0\) and \(t=1\) and a space singularity at \(x=0\). Moreover, \(g\) may be superlinear in its second variable \(x\) for \(x\to \infty\) and sublinear in its third variable \(y\) for \(| y| \to \infty\). The existence of a solution \(u\in C[0,1]\cup C^2(0,1)\) such that \(u(t)>0\) for \(t\in (0,1)\) and \(t(1-t)u'\in C[0,1]\) is proved. The proof is based on the Leray-Schauder fixed-point theorem and on the lower and upper solutions method. The author was motivated by P. Habets and F. Zanolin [J. Math. Anal. Appl. 181 No.3, 684-700 (1994; Zbl 0801.34029)] and by R.P. Agarwal and D. O’Regan [J. Differ. Equations 143 No.1, 60-95 (1998; Zbl 0902.34015)].

MSC:

34B16 Singular nonlinear boundary value problems for ordinary differential equations
34B18 Positive solutions to nonlinear boundary value problems for ordinary differential equations
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