论文标题
非局部相互作用的最小化物具有外源性的功能
Minimizers of nonlocal interaction functional with exogenous potential
论文作者
论文摘要
本文的目的是考虑以下非局部交互功能\ begin {qore*} e [ρ] = \ frac {1} {2} {2} \ int _ {\ mathbb {r}^n} \ int _ {\ mathbb {r}^n} k（x-y）ρ（x）ρ（y）dxdy+\ int _ {\ mathbb {r}^n} f（x）ρ（x）ρ（x）dx。 \ end {equation*} kernel $ k（x）= \ frac {1} {q} | x | x |^q- \ frac {1} {p} {p} | x |^p $是内源性的潜力，其中$ q> p> p> -n $。外源性潜在$ f $是一种非负连续功能，满足$ f（x）\ to +\ infty $作为$ | x | \ to +\ to +\ infty $。最小化的存在是根据浓度紧凑原理建立的。 Especially, for $F(x)=β|x|^2(β>0)$ and $K(x)=\frac{1}{2}|x|^2-\frac{1}{2-N}|x|^{2-N}$($N>2$), the global minimizer is given explicitly by the method of calculus of variation.
The purpose of this paper is to consider the minimization problem of the following nonlocal interaction functional \begin{equation*} E[ρ]=\frac{1}{2}\int_{\mathbb{R}^N} \int_{\mathbb{R}^N}K(x-y)ρ(x)ρ(y)dxdy+\int_{\mathbb{R}^N}F(x)ρ(x)dx. \end{equation*} The kernel $K(x)=\frac{1}{q}|x|^q-\frac{1}{p}|x|^p$ is an endogenous potential, where $q>p>-N$. The exogenous potential $F$ is a nonnegative continuous function and satisfies $F(x)\to +\infty$ as $|x|\to +\infty$. The existence of minimizers are established based on the concentration compactness principle. Especially, for $F(x)=β|x|^2(β>0)$ and $K(x)=\frac{1}{2}|x|^2-\frac{1}{2-N}|x|^{2-N}$($N>2$), the global minimizer is given explicitly by the method of calculus of variation.
