Multi-bump bound states for a system of nonlinear Schrödinger equations

Motivated by problems arising in nonlinear optics and Bose–Einstein condensates, we consider in RNRN, with N⩽3N⩽3, the following system of coupled Schrödinger equations:{−Δui+λVi(x)ui=ui∑j=1dαijuj2,ui⩾0,lim|x|→∞ui(x)=0,i=1,…,d, where λ>0λ>0 is a parameter, αij=αjiαij=αji are positive constants, and ViVi non-negative given potentials. We assume that the interior of ⋂i=1dVi−1(0) admits m   connected components Ω1,…,ΩmΩ1,…,Ωm which are of class C1C1, and isolated in each Vi−1(0). For each non-empty J⊂{1,…,m}J⊂{1,…,m}, we prove that the system admits for any λ   large a multi-bump solution uλ:RN→Rduλ:RN→Rd which is small in RN∖⋃j∈JΩjRN∖⋃j∈JΩj, and on each ΩjΩj (j∈Jj∈J) close in H1H1-norm to a least energy solution of the limit problem:−Δui=ui∑j=1dαijuj2,i=1,…,d, subjected to homogeneous Dirichlet boundary condition. An explicit condition on the matrix (αij)(αij) is given to ensure our solutions have at least two positive components.