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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 ' ),u(0)=u(1)=0·(1)

The nonlineearity g:(0,1)×(0,)× 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 and sublinear in its third variable y for |y|. The existence of a solution uC[0,1]C 2 (0,1) such that u(t)>0 for t(0,1) and t(1-t)u ' 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)].

34B16Singular nonlinear boundary value problems for ODE
34B18Positive solutions of nonlinear boundary value problems for ODE