Existence of τ-antisymmetric solutions for a system in RN

In this paper we prove the existence of a nontrivial τ-antisymmetric solution for the following system{−Δu+u=|u|2p−2u+β(x)|v|p|u|p−2u,inRN−Δv+ω2v=|v|2p−2v+β(x)|u|p|v|p−2v,inRN where N≥3, 2<2p<2⁎, ω>0 is a positive constant and τ:RN→RN is a nontrivial orthogonal involution. Here β is a continuous function, invariant under τ, satisfying additional conditions, some of them related to ω. The basic tool employed here is the Concentration Compactness Principle.