Existence of positive solution for a system of elliptic equations via bifurcation theory

In this paper we study the existence of solution for the following class of elliptic systems

(P)$\{\begin{array}{c}?\mathrm{\Delta }u=(a?{\int }_{\mathrm{\Omega }}K(x,y)f(u,v)dy)u+bv,\text{in}\mathrm{\Omega }\hfill \\ ?\mathrm{\Delta }v=(d?{\int }_{\mathrm{\Omega }}\mathrm{\Gamma }(x,y)g(u,v)dy)v+cu,\text{in}\mathrm{\Omega }\hfill \\ u=v=0,\text{on}?\mathrm{\Omega }\hfill \end{array}$

where $\mathrm{\Omega }?{\mathbb{R}}^N$ is a smooth bounded domain, $N\ge 1$, and $K,\mathrm{\Gamma }:\mathrm{\Omega }\times \mathrm{\Omega }\to \mathbb{R}$ are nonnegative functions satisfying some hypotheses and $a,b,c,d\in \mathbb{R}$. The functions f and g satisfy some conditions which permit to use Bifurcation Theory to prove the existence of solution for (P).