Population models involving the p-Laplacian with indefinite weight and constant yield harvesting

We consider the following reaction-diffusion equation-Δpu=am(x)up-1-buγ-1-ch(x),x∈Ω,u(x)=0,x∈∂Ω,where Δp denotes the p-Laplacian operator defined by Δpz = div(∣∇z∣p−2∇z); p > 1, γ(>p); a, b and c are positive constant, Ω is a smooth bounded domain in RN(N ⩾ 3) with ∂Ω of class C1,β for β ∈ (0, 1) and connected. The weight m satisfying m ∈ C(Ω) and m(x) ⩾ m0 > 0 for x ∈ Ω, also ∥m∥∞ = l < ∞ and h:Ω¯→R is a C1,α(Ω¯) function satisfying h(x) ⩾ 0 for x ∈ Ω, h(x) ≢ 0, maxh(x) = 1 for x∈Ω¯ and h(x) = 0 for x ∈ ∂Ω. Here u is the population density, am(x)up−1 − buγ−1 represents the logistic growth and ch(x) represents the constant yield harvesting rate [Oruganti S, Shi J, Shivaji R. Diffusive logistic equation with constant yield harvesting. I: steady states. Trans Am Math Soc 2002;354(9):3601–19]. We prove the existence of the positive solution under certain conditions.