Boundary Value Problems
Volume 2005 (2005), Issue 3, Pages 323-327
doi:10.1155/BVP.2005.323

Existence of a positive solution for a p-Laplacian semipositone problem

Maya Chhetri1 and R. Shivaji2

1Department of Mathematical Sciences, University of North Carolina at Greensboro, 27402, NC, USA
2Department of Mathematics and Statistics, Mississippi State University, Mississippi State, 39762, MS, USA

Received 30 September 2004; Revised 13 January 2005

Copyright © 2005 Maya Chhetri and R. Shivaji. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We consider the boundary value problem −Δpu=λf(u) in Ω satisfying u=0 on ∂Ω, where u=0 on ∂Ω, λ>0 is a parameter, Ω is a bounded domain in ℝn with C2 boundary ∂Ω, and Δpu:=div(|∇u|p−2∇u) for p>1. Here, f:[0,r]→ℝ is a C1 nondecreasing function for some r>0 satisfying f(0)<0 (semipositone). We establish a range of λ for which the above problem has a positive solution when f satisfies certain additional conditions. We employ the method of subsuper solutions to obtain the result.