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

Received 22 September 2005; Accepted 10 November 2005

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.