Article Contents
Article Contents

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

• *Corresponding author
• 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.

 Citation:

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