Multiplicity of positive solutions for a mixed boundary elliptic system

In this paper we are concerned with the existence and multiplicity of positive solutions for the following class of elliptic system {-epsilon(2)Delta u + u = Q(u)(u,v), -epsilon(2)Delta v + v = Q(v) (u,v) in Omega u = v = 0 on Gamma and partial derivative u/partial derivative eta = partial derivative v/partial derivative eta = 0 on Sigma where Omega is a bounded domain in R-N, Q is a p-homogeneous function with 2 < p < 2N/(N - 2) for N >= 3. The main tool used in this paper is the variational method combined with the Ljusternick-Schnirelman category of Sigma in itself.