Sign-changing solutions for coupled nonlinear Schrödinger equations with critical growth

We consider the following nonlinear Schrödinger system with critical growth−Δuj=λjuj+∑i=1kβij|ui|2⁎2|uj|2⁎2−2uj,inΩ,uj=0,on∂Ω,j=1,⋯,k, where Ω is a bounded smooth domain in RNRN, 2⁎=2NN−2, 0<λj<λ1(Ω)0<λj<λ1(Ω), j=1,⋯,kj=1,⋯,k, λ1(Ω)λ1(Ω) is the first eigenvalue of −Δ with zero Dirichlet boundary condition. We consider the repulsive case, namely βjj>0βjj>0, j=1,⋯,kj=1,⋯,k, βij=βji≤0βij=βji≤0, i≠ji≠j, i,j=1,⋯,ki,j=1,⋯,k. The existence of infinitely many sign-changing solutions as bound states is proved, provided N≥7N≥7, by approximations of systems with subcritical growth and by the concentration analysis on approximating solutions.