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 system of elliptic equations $$   \left\{   \begin{array}{lcl}   -\Delta u=\left(a-\int_{\Omega}K(x,y)f(u,v)dy\right)u+bv,\quad \mbox{in} \quad \Omega   -\Delta v=\left(d-\int_{\Omega}\Gamma(x,y)g(u,v)dy\right)v+cu,\quad \mbox{in} \quad \Omega u=v=0,\quad \mbox{on} \quad \partial\Omega   \end{array}   \right.   \eqno{(P)}   $$ where $\Omega\subset\R^N$ is a smooth bounded domain, $N\geq1$, and $K,\Gamma:\Omega\times\Omega\rightarrow\R$ is a nonnegative function checking some hypotheses and $a,b,c,d\in\R$. The functions $f$ and $g$ satisfy some conditions which permit to use Bifurcation Theory to prove the existence of solution for $(P)$. 
Romildo N. de Lima and Marco A. S. Souto
math.AP