Existence of solution for nonlinear integral equations of Hxc3xa9non type in critical case Abstract   In this paper, we will show that there exists a       xcexb4    xe2x88x97    u003e  0     such that when     xcexb1  u003e  0  ,  xcexb2  u003e  0     and     xcexb1    xcexb2      xcexb4    xe2x88x97      , the following problem    (0.1)       f      q    xcexb3    xe2x88x92  1     (  x  )       xe2x88xab    xcexa9           x       xcexb1         y       xcexb2    f   (  y  )          x  xe2x88x92  y       n  xe2x88x92  xcexb3        d  y    xcexbb    xe2x88xab    xcexa9      f   (  y  )          x  xe2x88x92  y       n  xe2x88x92  xcexb3  xe2x88x92  1        d  y  ,    x  xe2x88x88    xcexa9   xc2xaf       has at least one positive solution     f  xe2x88x88  C   (    xcexa9        )     , where    xcexa9    is the unit ball in       R    n     (  n  xe2x89xa5  3  )      centered at the origin,     1    xcexb3    n    ,     xcexbb  u003e  0     and       q    xcexb3    xe2x89x94    2  n    n    xcexb3      .

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.