On a p-Kirchhoff-type problem arising in ecosystems

Abstract

In this article, we discuss the existence of positive solutions for an ecological model of the form: $$\begin{aligned} \left\{ \begin{array}{ll} - M\left( \int _{\Omega }\mid \nabla u\mid ^{p} \mathrm{d}x\right) \Delta _{p} u = \frac{au^{p-1} - bu^{\gamma -1} - c}{u^{\alpha }}, &{}\quad x\in \Omega u= 0 , &{}\quad x\in \partial \Omega , \end{array}\right. \end{aligned}$$ where \(\Omega \) is a bounded domain with smooth boundary, \(\Delta _{p} u={\text {div}} (|\nabla u|^{p-2}\nabla u),\)\(1 0,\)\(b >0,\)\(c \ge 0,\) and \(\alpha \in (0, 1).\) 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 \(a,b,\gamma \) and small values of c.