Multiple sign-changing and semi-nodal solutions for coupled Schrödinger equations

We study the following coupled Schrödinger equations which have appeared as several models from mathematical physics:{−Δu1+λ1u1=μ1u13+βu1u22,x∈Ω,−Δu2+λ2u2=μ2u23+βu12u2,x∈Ω,u1=u2=0on ∂Ω. Here Ω⊂RNΩ⊂RN (N=2,3N=2,3) is a smooth bounded domain, λ1,λ2λ1,λ2, μ1,μ2μ1,μ2 are all positive constants. We show that, for each k∈Nk∈N there exists βk>0βk>0 such that this system has at least k sign-changing solutions (i.e., both two components change sign) and k   semi-nodal solutions (i.e., one component changes sign and the other one is positive) for each fixed β∈(0,βk)β∈(0,βk).