Existence of ground state solutions for the planar axially symmetric Schrödinger-Poisson system

This paper is concerned with the following planar Schrödinger-Poisson system

$ \left\{ \begin{array}{ll} -\triangle u+V(x)u+\phi u = f(x,u), \ \ \ \ x\in { \mathbb{R} }^{2},\\ \triangle \phi = u^2, \ \ \ \ x\in { \mathbb{R} }^{2}, \end{array} \right. $

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 Poincaré, Anal. Non Linéaire 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( \mathbb{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.