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 $a\in C(\mathbb{ R}^{ 2})$ is a $\mathbb{ Z}^{ 2}$-periodic function with $\inf_{ \mathbb{ R}^{ 2}}a>0$, $\gamma>0$, $b\geq0$, $p\geq2$ 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 $p\geq2$ and $q\geq2p-2$ via variational methods. The auxiliary functional $J_{ 1}$ plays a key role in the cases $p\geq3$. We also prove the radial symmetry of positive solutions (up to translations) for $p\geq2$ and $q\geq 2$. 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 $p\geq2$ and $b\geq0$.
Categories: math.AP