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A note on singular nonlinear boundary value problems for the one-dimensional \(p\)-Laplacian. (English) Zbl 0981.34013

A singular boundary value problem for the one-dimensional \(p\)-Laplacian \[ (\varphi_p(y'))'+ f(t, y)= 0,\quad t\in (0,1),\quad y(0)= y(1)= 0, \] with \(\varphi_p(s)=|s|^{p- 2}s\), \(p> 1\), is considered. A sufficient condition for the existence of a positive bounded solution \(y(t)\in C^1[0, 1]\) such that \(\varphi_p(y')\in C(0,1)\) is given. The authors do not impose a monotone condition on \(f\), because of that they apply a fixed-point theorem in cones.

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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References:

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