An Existence Result for a Semipositone Problem with a Sign Changing Weight

where p > 0, c > 0, and λ > 0 are parameters and Ω is an open bounded region with boundary ∂Ω in class C2 in Rn for n ≥ 1. Here g :Ω→ R is a Cα function while h :Ω→ R is a nonnegative Cα function with ‖h‖∞ = 1. When p = 1, (1.1) arises in population dynamics where 1/λ is the diffusion coefficient and ch(x) represents the constant yield harvesting. In this case (p = 1), when g(x) is a positive constant, various results have been established in [4]. Here we focus on sign changing weight functions g. To precisely define our classes of weight functions, we first let λ1 > 0 be the principal eigenvalue and φ > 0 with ‖φ‖∞ = 1 the corresponding eigenfunction of −Δ with the Dirichlet boundary conditions. It is well known that ∂φ/∂η < 0 on ∂Ω where η is the unit outward normal. Hence there exists δ > 0, σ > 0, andm> 0 such that