Infinitely many solutions for the perturbed Bose–Einstein condensates system

We consider the perturbed nonlinear elliptic system as following {−Δuj+λjuj=∑i=12βijui2uj+αj(x),uj∈H01(Ω),j=1,2, where ΩΩ is a bounded smooth domain in RN(N≤3)RN(N≤3) and λj>−λ1(Ω)λj>−λ1(Ω), λ1(Ω)λ1(Ω) is the first eigenvalue of −Δ−Δ with the Dirichlet boundary condition; the perturbation terms αj∈L2(Ω)αj∈L2(Ω) for j=1,2j=1,2. Assume that the matrix (βij)i,j=1,2(βij)i,j=1,2 is either positive definite when N=1,2N=1,2 or anti-symmetric when N=3N=3. We show the existence of infinitely many solutions to this system. The results are also true for the mm-coupled system, i.e., i,j=1,…,m(m≥2)i,j=1,…,m(m≥2).