Solution of perturbed Schrödinger system with critical nonlinearity and electromagnetic fields

In this paper, we consider the following perturbed nonlinear Schrödinger system with electromagnetic fields−( ϵ ∇ + iA ( x ) )2 u + V ( x ) u = H s( | u | 2 , | v | 2) u + K ( x ) | u |2 蜧 − 2 u , x ∈ R N ,−( ϵ ∇ + iA ( x ) )2 v + V ( x ) v = H t( | u | 2 , | v | 2) v + K ( x ) | v |2 蜧 − 2 v , x ∈ R N ,where N  ≥ 3, 2  蜧   = 2 N  ∕ ( N  − 2) is the Sobolev critical exponent; A is the real vector magnetic potential; and V ( x ), K ( x ), and H ( s , t ) are continuous functions. Under certain conditions on V , H , and K , we establish some new results on the existence of the least‐energy solutions ( u ϵ , v ϵ ) for small ϵ by using variational method. Copyright © 2012 John Wiley & Sons, Ltd.