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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

${u}^{\text{'}\text{'}}=g\left(t,u,{u}^{\text{'}}\right),\phantom{\rule{1.em}{0ex}}u\left(0\right)=u\left(1\right)=0·\phantom{\rule{2.em}{0ex}}\left(1\right)$

The nonlineearity $g:\left(0,1\right)×\left(0,\infty \right)×ℝ\to ℝ$ 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 $u\in C\left[0,1\right]\cup {C}^{2}\left(0,1\right)$ such that $u\left(t\right)>0$ for $t\in \left(0,1\right)$ and $t\left(1-t\right){u}^{\text{'}}\in C\left[0,1\right]$ 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 ODE 34B18 Positive solutions of nonlinear boundary value problems for ODE
##### Keywords:
singular BVP; upper and lower solutions; existence; superlinear