论文标题
Chern-Simons-Schrödinger系统的多峰解决方案的存在和局部唯一性
Existence and local uniqueness of multi-peak solutions for the Chern-Simons-Schrödinger system
论文作者
论文摘要
在本文中，我们考虑Chern-simons-Schrödinger系统\ begin {equation} \ left \ {\ begin {aligned} ＆ - - \ varepsilon^{2}ΔU+v（x）u+（a_ {0}+a_ {1}^{2}+a_ {2}^{2}^{2}^{2}） = a_2 u^2，\ \ \ partial_ {2} a_ {0} = - a_ {1} u^{2}，\\＆\ partial_ {1} a_ {2} a_ {2} - \ partial_ {2} a_ {2} a__ {1} a_ {1} = - {1} \ partial_ {1} a_ {1}+\ partial_ {2} a_ {2} = 0，\\ \ end {aligned} \ right。 \ end {equation}其中$ p> 2，$ $ \ varepsilon> 0 $是参数，$ v：\ mathbb {r}^{2}^{2} \ rightarrow \ rightArrow \ mathbb {r} $是一个有界的连续函数。在$ V（x）$的一些温和假设下，我们显示了积极多峰解决方案的存在和局部唯一性。我们的方法主要使用有限维度还原方法，各种局部Pohozaev身份，爆炸分析和最大原则。由于$ a_ {0}，a_ {1} $和$ a_ {2}所涉及的非本地术语，我们必须获得一系列新的技术估算值。
In the present paper, we consider the Chern-Simons-Schrödinger system \begin{equation} \left\{ \begin{aligned} &-\varepsilon^{2}Δu+V(x)u+(A_{0}+A_{1}^{2}+A_{2}^{2})u=|u|^{p-2}u,\,\,\,\,x\in \mathbb{R}^2,\\ &\partial_1 A_0 = A_2 u^2,\ \partial_{2}A_{0}=-A_{1}u^{2},\\ &\partial_{1}A_{2}-\partial_{2}A_{1}=-\frac{1}{2}|u|^{2},\ \partial_{1}A_{1}+\partial_{2}A_{2}=0,\\ \end{aligned} \right. \end{equation} where $p>2,$ $\varepsilon>0$ is a parameter and $V:\mathbb{R}^{2}\rightarrow\mathbb{R}$ is a bounded continuous function. Under some mild assumptions on $V(x)$, we show the existence and local uniqueness of positive multi-peak solutions. Our methods mainly use the finite dimensional reduction method, various local Pohozaev identities, blow-up analysis and the maximum principle. Because of the nonlocal terms involved by $A_{0},A_{1}$ and $A_{2},$ we have to obtain a series of new and technical estimates.
