Existence and symmetry of solutions to 2-D Schr\"{o}dinger-Newton equations

In this paper, we consider the following 2-D Schr"{o}dinger-Newton equations \begin{eqnarray*} -\Delta u+a(x)u+\frac{\gamma}{2\pi}\left(\log(|\cdot|)|u|^p\right){|u|}^{p-2}u=b{|u|}^{q-2}u \qquad $\text{in}$ ,,, $\mathbb{R}$^{2}, \end{eqnarray} where $a\in C($\mathbb{R}$^{2})$isa$$\mathbb{Z}$^{2}$-periodicfunctionwith$\inf_{$\mathbb{R}$^{2}}a>0$,$\gamma>0$,$b\geq0$,$p\geq2$and$q\geq 2$. By using ideas from \cite{CW,DW,Stubbe}, under mild assumptions, we obtain existence of ground state solutions and mountain pass solutions to the above equations for $p\ge 2$ and $q\ge 2p-2$ via variational methods. The auxiliary functional $J_1$ plays a key role in the cases $p\ge 3$. We also prove the radial symmetry of positive solutions (up to translations) for $p\ge 2$ and $q\ge 2$. The corresponding results for planar Schr"{o}dinger-Poisson systems will also be obtained. Our theorems extend the results in \cite{CW,DW} from $p=2$ and $b=1$ to general $p\ge 2$ and $b\ge 0$.