Boundary Value Problems
Volume 2006 (2006), Article ID 87483, 7 pages
doi:10.1155/BVP/2006/87483

Existence results for classes of p-Laplacian semipositone equations

Shobha Oruganti1 and R. Shivaji2

1Department of Mathematics, School of Science, The Behrend College, Penn State Erie, Erie 16563, PA, USA
2Department of Mathematics and Statistics, Mississippi State University, Mississippi State 39762, MS, USA

Received 22 September 2005; Accepted 10 November 2005

Copyright © 2006 Shobha Oruganti 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 study positive C1(Ω¯) solutions to classes of boundary value problems of the form −Δpu=g(x,u,c) in Ω,u=0 on ∂Ω, where Δp denotes the p-Laplacian operator defined by Δpz:=div(|∇z|p−2∇z); p>1, c>0 is a parameter, Ω is a bounded domain in RN; N≥2 with ∂Ω of class C2 and connected (if N=1, we assume that Ω is a bounded open interval), and g(x,0,c)<0 for some x∈Ω (semipositone problems). In particular, we first study the case when g(x,u,c)=λf(u)−c where λ>0 is a parameter and f is a C1([0,∞)) function such that f(0)=0, f(u)>0 for 0<u<r and f(u)≤0 for u≥r. We establish positive constants c0(Ω,r) and λ*(Ω,r,c) such that the above equation has a positive solution when c≤c0 and λ≥λ∗. Next we study the case when g(x,u,c)=a(x)up−1−uγ−1−ch(x) (logistic equation with constant yield harvesting) where γ>p and a is a C1(Ω¯) function that is allowed to be negative near the boundary of Ω. Here h is a C1(Ω¯) function satisfying h(x)≥0 for x∈Ω, h(x)≢0, and maxx∈Ω¯h(x)=1. We establish a positive constant c1(Ω,a) such that the above equation has a positive solution when c<c1 Our proofs are based on subsuper solution techniques.