Existence and stability of standing waves for a coupled nonlinear schrödinger system

We study the existence and stability of the standing waves of two coupled Schrödinger equations with potentials |x|bi(bi ∈ ℝ, i=1, 2). Under suitable conditions on the growth of the nonlinear terms, we first establish the existence of standing waves of the Schrödinger system by solving a L2-normalized minimization problem, then prove that the set of all minimizers of this minimization problem, then prove that the set of all minimizers of this minimization problem is stable. Finally, we obtain the least energy solutions by the Nehari method and prove that the orbit sets of these least energy solutions are unstable, which generalizes the results of [11] where b1 = b2 = 2.