JournalszaaVol. 30, No. 3pp. 305–318

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
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Abstract

We study positive solutions to the singular boundary value problem

{Δpu=λf(u)uβ\mboxin Ωu=0\mboxon Ω,\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 Δpu=\Delta_p u = div (up2u)(|\nabla u|^{p-2}\nabla u), p>1,λ>0,β(0,1)p > 1, \lambda > 0, \beta \in (0,1) and Ω\Omega is a bounded domain in RN,N1.\mathbb{R}^{N}, N \geq 1. Here f: [0,)(0,)f:~[0, \infty)\rightarrow (0, \infty) is a continuous nondecreasing function such that limuf(u)uβ+p1=0.\lim_{u\rightarrow \infty} \frac{f(u)}{u^{\beta+p-1}}= 0. We establish the existence of multiple positive solutions for certain range of λ\lambda when ff satisfies certain additional assumptions. A simple model that will satisfy our hypotheses is f(u)=eαuα+uf(u)=e^{\frac{\alpha u}{\alpha+u}} for α1.\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.

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