Abstract. This paper concerns the existence of positive radial ground states of the time-independent Schrödinger system ⎧ ⎨where n = 1, 2, 3, λ j > 0 and µ j > 0 for j = 1, 2, and β > 0. A result from Sirakov, Comm. Math. Phys. 271 (2007), 199-221, is improved.Consider the time-independent Schrödinger system (1)where n = 1, 2, 3, λ j > 0 and µ j > 0 for j = 1, 2, and β > 0. Solutions (u 1 (x), u 2 (x)) of (1) correspond to standing wave solutions (e iλ 1 t u 1 (x), e iλ 2 t u 2 (x)) of the timedependent system of 2 coupled nonlinear Schrödinger equationsThe system (2) stems from many physical problems, especially in nonlinear optics and in the Hartree-Fock theory for Bose-Einstein condensates; see, for example, [1,5,8,9,10,11,12,20,23]. Recently, (1) has attracted tremendous attention and has been studied extensively from the point of view of physics (see [1,10,11] for instance) as well as mathematics (see [2,3,4,13,14,15,16,18,19,21,22,24,25]). Solutions of (1) correspond to critical points of the energy functional (1)