Note on ground states of a nonlinear Schrodinger system

We give a sufficient condition for the existence of positive radial ground states of the time-independent Schrodinger system {-Delta u(1) + lambda u(1) = mu(1)u(1)(3) + beta(12)u(2)(2)u(1) + beta(13)u(3)(2)u(1), in R-n, -Delta u(2) + lambda u(2) = beta(12)u(1)(2)u(2) + mu(2)u(2)(3) + beta(23)u(3)(2)u(2), in R-n, -Delta u(3) + lambda u(3) = beta(13)u(1)(2)u(3) + beta(23)u(2)(2)u(3) + mu(3)u(3)(3), in R-n, u(1)(x) -> 0, u(2)(x) -> 0, u(3)(x) -> 0, as |x| -> infinity, where n = 1,2,3, lambda > 0, mu j > 0 and beta ij > 0 (i < j) for i, j = 1,2,3. And in some special cases, our conditions are also necessary. (C) 2011 Elsevier Inc. All rights reserved.