A local mountain pass type result for a system of nonlinear Schrödinger equations

We consider a singular perturbation problem for a system of nonlinear Schrödinger equations:where N = 2, 3, μ₁, μ₂, β > 0 and V₁(x), V₂(x): R^N → (0, ∞) are positive continuous functions. We consider the case where the interaction β > 0 is relatively small and we define for P ε R^N the least energy level m(P) for non-trivial vector solutions of the rescaled "limit" problem: We assume that there exists an open bounded set Λ ⊂ R^N satisfying We show that (*) possesses a family of non-trivial vector positive solutions which concentrates-after extracting a subsequence e{open}_n → 0-to a point P₀ ε Λ with m(P₀) = inf_PεΛm(P). Moreover (v_1e{open}(x), v_2e{open}(x)) converges to a least energy non-trivial vector solution of (**) after a suitable rescaling.