Existence and uniqueness of positive solutions of nonlinear Schrödinger systems

We prove that the Schrodinger system where n = 1, 2, 3, N ≥ 2, λ 1 = λ 2 = … = 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{ β 11 , β 22 , …, β NN } or n = 1 and 0 β β 11 , β 22, …, β NN }.