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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, Eun Kyoung Lee and R. Shivaji

(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