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){−Δu=(a−∫ΩK(x,y)f(u,v)dy)u+bv,inΩ−Δv=(d−∫ΩΓ(x,y)g(u,v)dy)v+cu,inΩu=v=0,on∂Ω where Ω⊂RN is a smooth bounded domain, N≥1, and K,Γ:Ω×Ω→R are nonnegative functions satisfying some hypotheses and a,b,c,d∈R. The functions f and g satisfy some conditions which permit to use Bifurcation Theory to prove the existence of solution for (P).