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

We study the multiplicity solutions for the nonlinear elliptic system { − Δ u + λ u = | u | p − 1 u + λ μ v in Ω , − Δ v + λ v = | v | p − 1 v + λ μ u in Ω , u = v = 0 on ∂ Ω , where Ω is an exterior domain in R N , N ⩾ 3 . We will show that under some assumptions, the system has at least 3 cat Ω ¯ [ Ω ¯ , R N ∖ B ρ ¯ ( 0 ) ] solutions.