ABSTRACT

In this article, we study the Schrödinger–Newton equations $\begin{array}{rl}& -\mathrm{\Delta }u+2\pi \phi u=\mu u+|u{|}^{p}u,\mathrm{i}\mathrm{n}{\mathbb{R}}^2,\\ & \mathrm{\Delta }\phi =u^2,\mathrm{i}\mathrm{n}{\mathbb{R}}^2.\end{array}$ We prove the existence of positive spherically symmetric decreasing solutions for $p\in (0,2)$ by constrained minimization methods. For p = 2, by the Gagliardo–Nirenberg inequality, an elaborate estimate implies similar existence results. We also give the regularity, exponential decay and blow-up behavior of these solutions.

中文翻译：

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平面薛定谔-牛顿方程的最小化器

摘要

在本文中，我们研究薛定谔-牛顿方程$\begin{array}{rl}& -\mathrm{\Delta }你+2\pi \phi 你=\mu 你+|你{|}^p你,\mathrm{一世}\mathrm{n}{\mathbb{R}}^2,\\ & \mathrm{\Delta }\phi ={你}^2,\mathrm{一世}\mathrm{n}{\mathbb{R}}^2.\end{array}$我们证明了正球对称递减解的存在$p\in (0,2)$通过约束最小化方法。对于p  = 2，根据 Gagliardo-Nirenberg 不等式，精细的估计意味着相似的存在结果。我们还给出了这些解决方案的规律性、指数衰减和爆炸行为。