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: $$\begin{aligned} \left\{ \begin{array}{ll} - M\left( \int _{\Omega }\mid \nabla u\mid ^{p} \mathrm{d}x\right) \Delta _{p} u = \frac{au^{p-1} - bu^{\gamma -1} - c}{u^{\alpha }}, &{}\quad x\in \Omega ,\\ u= 0 , &{}\quad x\in \partial \Omega , \end{array}\right. \end{aligned}$$-M∫Ω∣∇u∣pdxΔpu=aup-1-buγ-1-cuα,x∈Ω,u=0,x∈∂Ω,where $$\Omega $$Ω is a bounded domain with smooth boundary, $$\Delta _{p} u={\text {div}} (|\nabla u|^{p-2}\nabla u),$$Δpu=div(|∇u|p-2∇u),$$1< p < \gamma ,$$10,$$a>0,$$b >0,$$b>0,$$c \ge 0,$$c≥0, and $$\alpha \in (0, 1).$$α∈(0,1). This model describes the steady states of a logistic growth model with grazing and constant yield harvesting. It also describes the dynamics of the fish population with natural predation and constant yield harvesting. We discuss the existence of a positive solution for given $$a,b,\gamma $$a,b,γ and small values of c.


Introduction
In this paper, we are interested in the existence of positive solutions for the p-Kirchhoff-type problems ÀM R X j ru j p dx À Á D p u ¼ au pÀ1 À bu cÀ1 À c u a ; x 2 X; where M : ½0; 1Þ À! ð0; 1Þ is a continuous and increasing function, c ! 0; a; b [ 0; X is a bounded domain with smooth boundary, D p denotes the p-Laplacian operator defined by D p z ¼ div ðjrzj pÀ2 rzÞ; 1\p\c and a 2 ð0; 1Þ.
Here u is the population density and au pÀ1 Àbu cÀ1 u a represents logistics growth. This model describes grazing of a fixed number of grazers on a logistically growing species (see [11]). The herbivore density is assumed to be a constant which is a valid assumption for managed grazing systems and the rate of grazing is given by c u a : At high levels of vegetation density this term saturates to c as the grazing population is a constant. This model has also been applied to describe the dynamics of fish populations (see [15]). In the case of the fish population the term c u a corresponds to natural predation. In recent years, problems involving Kirchhoff-type operators have been studied in many papers, we refer to [3,4,6,10,14] in which the authors have used the variational and topological methods to get the existence of solutions. In this article, we are motivated by the ideas introduced in [7,12,13] and properties of Kirchhoff-type operators in [3,4,6], we study problem (1) in semipositone case (i.e., lim sÀ!0 þ f ðsÞ ¼ À1; f ðsÞ ¼ as pÀ1 Àbs cÀ1 Àc s a Þ; see [5,[7][8][9]). Using sub-supersolution techniques, we prove the existence of a positive solution for the problem.
To precisely state our existence result we consider the eigenvalue problem Let / be the eigenfunction corresponding to the first eigenvalue k 1 of (3) such that /ðxÞ [ 0 in X and jj/jj 1 ¼ 1: It can be shown that o/ on \0 on oX: Here n is the outward normal. Let m; d [ 0 and l [ 0 be such that: with X d :¼ x 2 Xjdðx; oXÞ d f g : This is possible since jr/j p 6 ¼ 0 on oX while / ¼ 0 on oX. We will also consider the unique solution e 2 W 1;p 0 ðXÞ of the boundary value problem & to discuss our existence result, it is known that e [ 0 in X and oe on \0 on oX:

Existence results
In this section, we shall establish our existence result via the method of sub-supersolution. A function w is said to be a subsolution of (1), if it is in W 1;p 0 ðXÞ such that and z is said supersolution of (1), if it is in W 1;p 0 ðXÞ such that Then the following result holds: Then the following result holds: (1) has a positive solution.

Remark 2.3
In the nonsingular case ða ¼ 0Þ, positive solutions exist only when a [ k 1 (the principle eigenvalue) (see [12,13]). But in the singular case, we establish the existence of a positive solution for any a [ 0.
Proof of Theorem 2.2 We start with the construction of a positive subsolution for (1). Fix b 2 ð1; p pÀ1þa Þ: Define : Note that c 1 [ 0 by the choice of k and b. A calculation shows that À ðb À 1Þðp À 1Þjr/j p / ÀpþbðpÀ1Þ wdx: Thus w is a subsolution of (1) if M 1 k pÀ1 b pÀ1 k 1 / bðpÀ1Þ À ðb À 1Þðp À 1Þjr/j p / ÀpþbðpÀ1Þ h i ak pÀ1Àa / bðpÀ1ÀaÞ À bk cÀ1Àa / bðcÀ1ÀaÞ À c k a / ab : For this, we have to show the following three inequalities: À k pÀ1Àa / bðpÀ1ÀaÞ a À M 1 k a b pÀ1 k 1 / ab À Á À 2bk cÀ1Àa / bðcÀ1ÀaÞ ; x 2 X; by the choice of k, we have: Now, we have in X d ; jr/j p ! m; and c\M 1 k pÀ1þa b pÀ1 ðb À 1Þðp À 1Þm p ; then the following inequalities hold: On the other hand, since p À bðp À 1 þ aÞ [ 0; À c k a / ab / pÀbðpÀ1ÞÀab À c k a / ab : Finally, in X À X d using / ! l and c\ 1 2 M 1 k pÀ1 l bðpÀ1Þ ða À b pÀ1 k 1 k a Þ, we have: For c\c 1 , by (6) and (7) the Eq. (5) holds. Thus w is a subsolution of (1). Now for a supersolution choose z :¼ Ne; where N [ 0 is such that Ne ! w and au pÀ1 À bu cÀ1 À c M 0 u a N pÀ1 ; for all u [ 0: We have ÀM Z X j rz j p dx i.e., z is a supersolution of (1) with z ! w for N large (note jrej 6 ¼ 0; oXÞ: Thus, there exists a positive solution u of (1) such that w u z: This completes the proof of Theorem 2.2. h