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二维Schrödinger-Newton方程解的存在性和对称性

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

Daomin Cao, Wei Dai, Yang Zhang

发表信息：
Dynamics of Partial Differential Equations, 18 (2021), no. 2, 113-156
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摘要：

本文考虑了二维薛定谔-牛顿方程\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*}，其中$a\in C(\mathbb{R}^{2})$是具有$\inf_{\mathbb{R}^{2}}a>0$，$γ>0$，$b\geq0$，$p\geq2$和$q\geq 2$的$\mathbb{Z}^{2}$-周期函数.利用\cite{CW,DW,Stubbe}的思想，在较温和的假设下，利用变分方法得到了上述方程的$p\geq2$和$q\geq2p-2$的基态解和山路解的存在性.辅助功能$J_{1}$在$p\geq3$病例中起关键作用。我们还证明了$p\geq2$和$q\geq 2$正解的径向对称性（直到平移）.对于平面Schrödinger-Poisson系统也将得到相应的结果。我们的定理将\cite{CW,DW}中的结果从$p=2$和$b=1$推广到一般的$p\geq2$和$b\geq0$.

英文摘要：

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 $b\geq0$.

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