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Positive solutions for the one-dimensional singular superlinear $ p $-Laplacian problem

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  • We prove the existence of positive classical solutions for the $ p $-Laplacian problem

    $ \begin{equation*} \left\{ \begin{array}{c} -(r(t)\phi (u^{\prime }))^{\prime } = -\frac{\lambda }{u^{\delta }}+f(t,u),\ t\in (0,1), \\ u(0) = u(1) = 0,\end{array}\right. \end{equation*} $

    where $ 0<\delta <1 $, $ \phi (s) = |s|^{p-2}s $, $ p>1 $, $ f:(0,1)\times \lbrack 0,\infty )\rightarrow \mathbb{R} $ is a Carathéodory function satisfying $ \limsup\limits_{z\rightarrow 0^{+}}\frac{f(t,z)}{z^{p-1}}<\lambda _{1}<\liminf\limits_{z\rightarrow \infty }\frac{f(t,z)}{z^{p-1}} $ uniformly for a.e. $ t $ $ \in (0,1), $ where $ \lambda_{1} $ denotes the principal eigenvalue of $ -(r(t)\phi (u^{\prime }))^{\prime } $ with zero boundary conditions, and $ \lambda $ is a small nonnegative parameter.

    Mathematics Subject Classification: 34B16, 34B18.


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