论文标题
2-DSchrödinger-Newton方程的解决方案的存在和对称性
Existence and symmetry of solutions to 2-D Schrödinger-Newton equations
论文作者
论文摘要
在本文中，我们考虑以下2-DSchrödinger-Newton方程\ begin {eqnarray*}-ΔU+a（x）u+++\fracγ{2π} \ left（\ log（| \ cdot |）*| \ text {in} \，\，\，\ Mathbb {r}^{2}，\ end {eqnarray*}其中$ a \ in c（\ mathbb {r}^{2}）$ is a $ \ \ sathbb {z} $ \ inf _ {\ mathbb {r}^{2}} a> 0 $，$γ> 0 $，$ b \ geq0 $，$ p \ geq2 $和$ q \ geq 2 $。通过使用\ cite {cw，dw，stubbe}的想法，在温和的假设下，我们通过变异方法获得了$ p \ geq2 $和$ p \ geq2 $和$ p \ geq2 $和$ p \ geq2p-2 $的基态解决方案和山间通道解决方案。辅助功能$ j_ {1} $在$ p \ geq3 $的情况下扮演关键角色。我们还证明了$ p \ geq2 $和$ q \ geq 2 $的阳性解决方案的径向对称性（升至翻译）。还将获得平面Schrödinger-Poisson系统的相应结果。我们的定理将结果从$ p = 2 $和$ b = 1 $扩展到\ cw，dw}的结果，向常规$ p \ geq2 $和$ b \ geq0 $。
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 $\inf_{\mathbb{R}^{2}}a>0$, $γ>0$, $b\geq0$, $p\geq2$ 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 Schrödinger-Poisson systems will also be obtained. Our theorems extend the results in \cite{CW,DW} from $p=2$ and $b=1$ to general $p\geq2$ and $b\geq0$.
