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Zeitschrift für Analysis und ihre Anwendungen


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Volume 30, Issue 3, 2011, pp. 305–318
DOI: 10.4171/ZAA/1436

Published online: 2011-07-03

Multiplicity Results for Classes of Infinite Positone Problems

Eunkyung Ko[1], Eun Kyoung Lee[2] and R. Shivaji[3]

(1) Mississippi State University, USA
(2) Pusan National University, Busan, South Korea
(3) Mississippi State University, USA

We study positive solutions to the singular boundary value problem \begin{equation*}\left\{\begin{alignedat}{2}-\Delta_p u &= \lambda \frac{f(u)}{u^\beta}& \quad &\mbox{in}~\Omega \\ u &= 0 & \quad &\mbox{on}~\partial \Omega, \end{alignedat}\right. \end{equation*} where $\Delta_p u =$ div $(|\nabla u|^{p-2}\nabla u)$, $ p > 1, \lambda > 0, \beta \in (0,1)$ and $ \Omega$ is a bounded domain in $\mathbb{R}^{N}, N \geq 1.$ Here $f:~[0, \infty)\rightarrow (0, \infty)$ is a continuous nondecreasing function such that $\lim_{u\rightarrow \infty} \frac{f(u)}{u^{\beta+p-1}}= 0.$ We establish the existence of multiple positive solutions for certain range of $\lambda$ when $f$ satisfies certain additional assumptions. A simple model that will satisfy our hypotheses is $f(u)=e^{\frac{\alpha u}{\alpha+u}}$ for $\alpha \gg 1.$ We also extend our results to classes of systems when the nonlinearities satisfy a combined sublinear condition at infinity. We prove our results by the method of sub-supersolutions.

Keywords: Singular boundary value problems, infinite positone problems, multiplicity of positive solutions, sub-supersolutions

Ko Eunkyung, Lee Eun Kyoung, Shivaji R.: Multiplicity Results for Classes of Infinite Positone Problems. Z. Anal. Anwend. 30 (2011), 305-318. doi: 10.4171/ZAA/1436