Existence results for classes of p-Laplacian semipositone equations, Boundary Value Problems

We study positive C1(Ω̄) 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) = λ f (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(Ω̄) function that is allowed to be negative near the boundary of Ω. Here h is a C1(Ω̄) function satisfying h(x) ≥ 0 for x ∈ Ω, h(x) ≡ 0, and maxx∈Ω̄h(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. Copyright © 2006 S. Oruganti and R. Shivaji. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, dis-tribution, and reproduction in any medium, provided the original work is properly cited. 1