# Multiplicity Results for Classes of Infinite Positone Problems

### Eunkyung Ko

Mississippi State University, USA### Eun Kyoung Lee

Pusan National University, Busan, South Korea### R. Shivaji

Mississippi State University, USA

## Abstract

We study positive solutions to the singular boundary value problem

\[ \left\{\begin{alignedat}{2}-\Delta_p u &= \lambda \frac{f(u)}{u^\beta}& \quad &\mbox{in}~\Omega \\ u &= 0 & \quad &\mbox{on}~\partial \Omega, \end{alignedat}\right. \]where $Δ_{p}u=$ div $(∣∇u∣_{p−2}∇u)$, $p>1,λ>0,β∈(0,1)$ and $Ω$ is a bounded domain in $R_{N},N≥1.$ Here $f:[0,∞)→(0,∞)$ is a continuous nondecreasing function such that $lim_{u→∞}u_{β+p−1}f(u) =0.$ We establish the existence of multiple positive solutions for certain range of $λ$ when $f$ satisfies certain additional assumptions. A simple model that will satisfy our hypotheses is $f(u)=e_{α+uαu}$ for $α≫1.$ 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.

## Cite this article

Eunkyung Ko, Eun Kyoung Lee, R. Shivaji, Multiplicity Results for Classes of Infinite Positone Problems. Z. Anal. Anwend. 30 (2011), no. 3, pp. 305–318

DOI 10.4171/ZAA/1436