Existence and uniqueness of positive solutions of nonlinear Schrödinger systems

Abstract

We prove that the Schrödinger system

where n = 1, 2, 3, N ≥ 2, λ₁ = λ₂ = … = λ_N = 1, β_ij = β_ji > 0 for i, j = 1, …, N, has a unique positive solution up to translation if the β_ij (i ≠ j) are comparatively large with respect to the β_jj. The same conclusion holds if n = 1 and if the β_ij (i ≠ j) are comparatively small with respect to the β_jj. Moreover, this solution is a ground state in the sense that it has the least energy among all non-zero solutions provided that the β_ij (i ≠ j) are comparatively large with respect to the β_jj, and it has the least energy among all non-trivial solutions provided that n = 1 and the β_ij (i ≠ j) are comparatively small with respect to the β_jj. In particular, these conclusions hold if β_ij = (i ≠ j) for some β and either β > max{β₁₁, β₂₂, …, β_NN} or n = 1 and 0 < β < min{β₁₁, β22, …, β_NN}.