Stability and instability of standing waves to a system of Schrِdinger equations with combined power-type nonlinearities

We obtain the existence of standing wave solutions to a coupled nonlinear Schrödinger system { − i u 1 t = Δ u 1 + ( μ | u 1 | p + ∑ k = 1 m α k | u 1 | p k − 2 | u 2 | q k ) u 1 in R N × ( 0 , T ) , − i u 2 t = Δ u 2 + ( γ | u 2 | p + ∑ k = 1 m β k | u 1 | p k | u 2 | q k − 2 ) u 2 in R N × ( 0 , T ) , u 1 ( x , 0 ) = u 10 ( x ) , u 2 ( x , 0 ) = u 20 ( x ) in R N , where 0 < p < 4 ( N − 2 ) + , μ, γ and α k , β k , p k , q k are real constants with 1 < p k < p + 1 , 1 < q k < p + 1 , p k + q k = p + 2 and α k q k = β k p k ( k = 1 , … , m ) . We also discuss the stability and instability of the standing wave solutions.