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\mbox{ 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\mbox{ in }\Omega,\\ \qquad \qquad \qquad \qquad u=v&=&0\mbox{ 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 problem $(P)$. 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.