Axially symmetric solutions of the Schrxc3xb6dingerxe2x80x93Poisson system with zero mass potential in R2 Abstract   In this paper, we consider the following planar Schrodingerxe2x80x93Poisson system           xe2x88x92  xcex94  u    xcfx95  u    K   (  x  )   f   (  u  )   ,          x  xe2x88x88    R    2    ,      xcex94  xcfx95      u    2    ,          x  xe2x88x88    R    2    ,          where     K  xe2x88x88  C   (    R    2    ,   (  0  ,  xe2x88x9e  )   )      and     f  xe2x88x88  C   (  R  ,  R  )     . By constructing a new energy estimate inequality and applying some new tricks, we prove that the above system has an axially symmetric solution which has a special minimax characterisation. Our result extends previous results in the literature.

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.