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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 $$ u''= g(t,u,u'),\quad u(0)=u(1)=0. \tag1$$ The nonlineearity $g: (0,1)\times (0,\infty)\times \bbfR \to \bbfR$ 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 $\vert y\vert \to \infty$. 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 {\it P. Habets} and {\it F. Zanolin} [J. Math. Anal. Appl. 181 No.3, 684-700 (1994; Zbl 0801.34029)] and by {\it R.P. Agarwal} and {\it D. O’Regan} [J. Differ. Equations 143 No.1, 60-95 (1998; Zbl 0902.34015)].

MSC:
34B16Singular nonlinear boundary value problems for ODE
34B18Positive solutions of nonlinear boundary value problems for ODE
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References:
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