On the existence of a positive solution to a nonlocal logistic system with nonlinear advection terms

In this paper, we study a nonlocal logistic system with nonlinear advection terms$ \begin{equation*} \left\{ \begin{array}{lcl} -\Delta u+\vec{\alpha}(x)\cdot \nabla (|u|^{p-1}u)& = &\left(a-\int_{\Omega}K_1(x, y)f(u, v)dy \right)u+bv\text{ in }\Omega, \\ -\Delta v+\vec{\beta}(x)\cdot \nabla (|v|^{q-1}v)& = &\left(d-\int_{\Omega}K_2(x, y)g(u, v)dy \right)v+cu\text{ in }\Omega, \\ \qquad \qquad \qquad \qquad u = v& = &0\text{ on }\partial\Omega, \end{array} \right. \end{equation*} $where $ \Omega\subset \mathbb{R}^N $, $ N\geq1 $, is a bounded domain with a smooth boundary, $ \vec{\alpha}(x) = (\alpha_1(x), \cdots, \alpha_N(x)) $ and $ \vec{\beta}(x) = (\beta_1(x), \cdots, \beta_N(x)) $ are flows satisfying suitable conditions, $ p, q\geq1 $, $ a, b, c, d>0 $ and $ K_1, K_2:\Omega\times\Omega\rightarrow \mathbb{R} $ are nonnegative functions, with their specific conditions detailed below. The functions $ f $ and $ g $ satisfy some assumptions which allow us to use bifurcation theory to prove the existence of solution to this problem. It is important to highlight that the inclusion of the integral nonlocal term on the right-hand side makes the problem more representative of real-world situations.