论文标题
具有局部潜力的Schrödinger-Poisson系统解决方案的存在和浓度
Existence and concentration of solution for Schrödinger-Poisson system with local potential
论文作者
论文摘要
在本文中，我们研究以下非线性schrödinger -Poisson类型方程\ begin {qore*} \ begin {cases} - \ varepsilon^2Δu+v（x）u+k（x） - \ \ varepsilon^2δϕ = k（x）u^2＆\ text {in} \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ {r}^3，\ end {cases} \ end {equepation {equation*}，其中$ \ varepsilon> 0 $是一个小参数，$ v：\ mathbb { $ k：\ mathbb {r}^3 \ rightarrow \ mathbb {r} $用于描述电子费用。在$ v（x），k（x）$和$ f $上的合适假设下，我们证明了$ \ varepsilon> 0 $ small的基态解决方案的存在和集中属性。此外，我们总结了Schrödinger-Poisson系统的一些开放问题。
In this paper, we study the following nonlinear Schrödinger-Poisson type equation \begin{equation*} \begin{cases} -\varepsilon^2Δu+V(x)u+K(x)ϕu=f(u)&\text{in}\ \mathbb{R}^3,\\ -\varepsilon^2Δϕ=K(x)u^2&\text{in}\ \mathbb{R}^3, \end{cases} \end{equation*} where $\varepsilon>0$ is a small parameter, $V: \mathbb{R}^3\rightarrow \mathbb{R}$ is a continuous potential and $K: \mathbb{R}^3\rightarrow \mathbb{R}$ is used to describe the electron charge. Under suitable assumptions on $V(x), K(x)$ and $f$, we prove existence and concentration properties of ground state solutions for $\varepsilon>0$ small. Moreover, we summarize some open problems for the Schrödinger-Poisson system.
