The effect of the domain topology on the number of positive solutions for an elliptic system

In this paper we prove an existence result of multiple positive solutions for the following system {-Delta u=2 alpha & varepsilon;/alpha & varepsilon;+beta & varepsilon;|u|(alpha & varepsilon;-2)u|v|(beta & varepsilon;) in Omega, -Delta v=2 beta & varepsilon;/alpha & varepsilon;+beta & varepsilon;|u|(alpha & varepsilon;)|v|(beta & varepsilon;-2)v in Omega, u=v=0 on partial derivative Omega, where Omega is a smooth and bounded domain in RN, N >= 3, alpha & varepsilon;=alpha-& varepsilon;/2, beta & varepsilon;=beta-& varepsilon;/2, alpha,beta>1 and alpha+beta=2(& lowast;), where 2(& lowast;)=2N/N-2. More specifically, we prove that, for & varepsilon;>0 small, the number of positive solutions is estimated below by topological invariants of the domain Omega: the Ljusternick-Schnirelmann category.