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

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.