Normalized vector solutions of nonlinear Schrödinger systems

Given μ>0 we look for solutions λ∈ℝ and v₁,…,vₖ ∈ H¹(ℝᴺ) of the system
−Δvᵢ+λvᵢ+Vᵢ(x)vᵢ=∑ⱼ₌₁ᵏj βᵢⱼ vᵢ vⱼ² and ∫_{ℝᴺ} (v₁² + ⋯ + vₖ²) dx = μ, in ℝᴺ, i=1,…,k,
where N=1,2,3, Vᵢ:ℝN→ℝ and βᵢⱼ∈ℝ satisfy βᵢⱼ=βⱼᵢ and βᵢᵢ>0. Under suitable assumptions on the βᵢⱼ's, given a non-degenerate critical point ξ₀ of a suitable linear combination of the potentials Vᵢ, we build solutions whose components concentrate at ξ₀ as the prescribed global mass μ is either large (when N=1) or small (when N=3) or it approaches some critical threshold (when N=2).