Multiple critical orbits to partial periodic perturbations of the prelativistic operator Abstract   We prove the existence of     m    1     geometrically distinct periodic solutions for    N   dimensional systems involving the    p   relativistic operator         R    p    u  xe2x89x94               u    xe2x80xb2         p  xe2x88x92  2      u    xe2x80xb2         (  1  xe2x88x92         u    xe2x80xb2         p    )     1  xe2x88x92  1  xe2x88x95  p          xe2x80xb2        and an    m   periodic potential; here     m    N    . This is obtained by reduction to an equivalent nonsingular problem for which the existence of multiple critical orbits can be inferred.

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.