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Zeitschrift für Analysis und ihre Anwendungen


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Volume 32, Issue 3, 2013, pp. 339–348
DOI: 10.4171/ZAA/1488

Published online: 2013-07-03

On a Singular Logistic Equation with the $p$-Laplacian

Dang Dinh Hai[1]

(1) Mississippi State University, USA

We prove the existence and nonexistence of positive solutions for the boundary value problems% \begin{equation*} \left\{ \begin{alignedat}{2} -\Delta _{p}u&= g(x,u)-\frac{h(x)}{u^{\alpha }}&\quad &\text{in }\Omega \\ u&= 0&&\text{on }\partial \Omega ,% \end{alignedat}% \right. \end{equation*}% where $\Delta _{p}u=\text{div}(|\nabla u|^{p-2}\nabla u),p>1,\ \Omega $ is a bounded domain in $\mathbb{R}^{n}$ with smooth boundary $\partial \Omega $, $% \alpha \in (0,1),g:\Omega \times (0,\infty )\rightarrow \mathbb{R}$ is possibly singular at $u=$ $0.\ $ An application to a singular logistic-like equation is given.

Keywords: Sup-supersolutions, singular, positive solutions

Hai Dang Dinh: On a Singular Logistic Equation with the $p$-Laplacian. Z. Anal. Anwend. 32 (2013), 339-348. doi: 10.4171/ZAA/1488