EXISTENCE OF GROUND STATE SOLUTIONS FOR THE PLANAR AXIALLY SYMMETRIC SCHRODINGER-POISSON SYSTEM

This paper is concerned with the following planar Schrodinger-Poisson system {-Delta u + V (x)u + phi u = f (x, u), x is an element of R-2, Delta phi = u(2), x is an element of R-2, where V (x) and f (x, u) are axially symmetric in x, and f (x, u) is asymptotically cubic or super-cubic in u. With a different variational approach used in [S. Cingolani, T. Weth, Ann. Inst. Henri Poincare, Anal. Non Lineaire 33 (2016) 169-197], we obtain the existence of an axially symmetric Nehari-type ground state solution and a nontrivial solution for the above system. The axial symmetry is more general than radial symmetry, but less used in the literature, since the embedding from the space of axially symmetric functions to L-s (R-N) is not compact. Our results generalize previous ones in the literature, and some of new phenomena do not occur in the corresponding problem for higher space dimensions.