Multiplicity solutions of an elliptic system in exterior domain with rich topology

We study the multiplicity solutions for the nonlinear elliptic system{−Δu+λu=|u|p−1u+λμvin Ω,−Δv+λv=|v|p−1v+λμuin Ω,u=v=0on ∂Ω, where Ω   is an exterior domain in RNRN, N⩾3N⩾3. We will show that under some assumptions, the system has at least 3catΩ¯[Ω¯,RN∖Bρ¯(0)] solutions.