On a p-Kirchhoff-type problem arising in ecosystems

In this article, we discuss the existence of positive solutions for an ecological model of the form: {-M(∫Ω∣∇u∣pdx)Δpu=aup-1-buγ-1-cuα,x∈Ω,u=0,x∈∂Ω,where Ω is a bounded domain with smooth boundary, Δ pu= div (| ∇ u| p-2∇ u) , 1 < p< γ, M: [0 , ∞) ⟶ (0 , ∞) is a continuous and increasing function, a> 0 , b> 0 , c≥ 0 , and α∈ (0 , 1). This model describes the steady states of a logistic growth model with grazing and constant yield harvesting. It also describes the dynamics of the fish population with natural predation and constant yield harvesting. We discuss the existence of a positive solution for given a, b, γ and small values of c. © 2015, The Author(s).