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In this paper, we consider the following weakly coupled nonlinear Schrödinger system−   ε  2   Δ   u  1   +   V  1    (  x  )    u  1    =  |    u  1    |   2  p  −  2u  1    +  β  |    u  1    |   p  −  2|   u  2   |   p    u  1   ,x  ∈   R  N   ,−   ε  2   Δ   u  2   +   V  2    (  x  )    u  2    =  |    u  2    |   2  p  −  2u  2    +  β  |    u  2    |   p  −  2|   u  1   |   p    u  2   ,x  ∈   R  N   ,where    ε  >  0   ,    β  ∈  R    is a coupling constant,    2  p  ∈  (  2  ,   2  ∗   )    with2  ∗   =    2  N    N  −  2if    N  ≥  3    and    +  ∞    if    N  =  1  ,  2   ,  V  1  and  V  2  belong to    C  (   R  N   ,   [  0  ,  ∞  )   )   .  When    p  ≥  2    and    β  >  0    is suitably small, we show that the problem has a family of nonstandard solutions    {   w  ε   =  (   u  ε  1   ,   u  ε  2   )  :  0  <  ε  <   ε  0   }    concentrating synchronously at the common local minimum of  V  1  and  V  2 . All decay rates ofV  i    (  i  =  1  ,  2  )are admissible and we can allow that    β  >  0    is close to 0 in this paper. Moreover, the location of concentration points is given by local Pohozaev identities. Our proofs are based on variational methods and the penalized technique.

Nonstandard solutions for a perturbed nonlinear Schrödinger system with small coupling coefficients