Existence and symmetry of solutions to 2-D Schrödinger-Newton equations

Abstract

In this paper, we consider the following 2-D Schrödinger-Newton equations \begin{eqnarray*} -Δu+a(x)u+\fracγ{2π}\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 ${\inf }_{{\mathbb{R}}^2}a>0$, $\gamma >0$, $b\ge 0$, $p\ge 2$ and $q\ge 2$. 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 Schrödinger-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 $b\ge 0$.

References in corpus (3)

• The one-dimensional Schrödinger-Newton equations
• Bound states of two-dimensional Schrödinger-Newton equations
• Ground states and high energy solutions of the planar Schrödinger-Poisson system

Cited by in corpus (1)

• Classification of solutions to conformally invariant systems with mixed order and exponentially increasing or nonlocal nonlinearity