Existence and asymptotic behavior of vector solutions for linearly coupled Choquardtype systems Abstract   We are concerned with the following linearly coupled Choquardtype system:           xe2x88x92  xcex94  u      xcexbb    1    u     (    I    xcexb1    xe2x88x97  F   (  u  )   )     F    xe2x80xb2     (  u  )     xcexb2  v      in      R    N    ,      xe2x88x92  xcex94  v      xcexbb    2    v     (    I    xcexb1    xe2x88x97  F   (  v  )   )     F    xe2x80xb2     (  v  )     xcexb2  u      in      R    N    ,          where     N  xe2xa9xbe  3    ,     xcexb1  xe2x88x88   (  0  ,  N  )   ,           xcexbb    i    u003e  0     are constants for     i    1  ,  2    ,     xcexb2  u003e  0     is a parameter,      I    xcexb1      is the Riesz potential, and     F  xe2x88x88    C    1     (  R  ,  R  )   .     Under some suitable conditions on       F    xe2x80xb2    ,     the existence of vector solutions is proved by using variational methods. Moreover, we prove the asymptotic behavior of the solutions as     xcexb2  xe2x86x92    0        .

Existence and concentration of ground state solutions for critical Schrxc3xb6dingerxe2x80x93Poisson system with steep potential well Abstract   In this paper, we consider the following Schrodingerxe2x80x93Poisson system                xe2x88x92   xcex94   u     (  1    xcexbc  g   (  x  )   )   u    xcfx95  u        u     4   u    xcexbb      u      q  xe2x88x92  2    u  ,        in      R  3   ,         xe2x88x92   xcex94   xcfx95     u  2   ,        in      R  3   ,             where qxe2x80xafxe2x88x88xe2x80xaf(3, 6) and xcexbb, xcexbcxe2x80xafu003exe2x80xaf0 are positive parameters. Since     f   (  u  )   :        u     4   u    xcexbb      u      q  xe2x88x92  2    u     with qxe2x80xafxe2x88x88xe2x80xaf(3, 4 does not satisfy the (AR) condition. Thus, we construct NehariPohoxc5xbeaevPalaisSmale sequence to overcome the boundedness of sequence. As qxe2x80xafxe2x88x88xe2x80xaf(4, 6), the boundedness of sequence is easily obtained. We need (g1) and (g2) to prove that      c  xcexbc      1  3    S   3  2       independent of xcexbc. Furthermore, we utilize the definition of the set of solutions to seek a ground state solution. Besides, the concentration behavior of the ground state solution is also described as xcexbcxe2x80xafxe2x86x92xe2x80xafxe2x88x9e.