Zeitschrift für angewandte Mathematik und Physik ( IF 2 ) Pub Date : 2021-05-02 , DOI: 10.1007/s00033-021-01541-7 Abderrazek Benhassine
This paper aims to study the existence and concentration of solutions for the stationary Dirac equation in \(\mathbb {R}^3\) with critical nonlinearities:
$$\begin{aligned} i\varepsilon \sum \limits _{k=1}^3\alpha _k\partial _k u-a\beta u+V(x)u=P(x)f(|u|)u+Q(x)g(|u|)u, \end{aligned}$$where \(\varepsilon >0\) is a small parameter and \(a>0\) is a constant. We also show the semiclassical solutions \(\omega _\varepsilon \) with maximum points \(x_\varepsilon \) concentrating at a special set \({\mathcal {H}}_{P}\) characterized by V(x), P(x) and Q(x), and for any sequence, \(x_\varepsilon \rightarrow x_0\in {\mathcal {H}}_{P}, v_\varepsilon (x):=\omega _\varepsilon (\varepsilon x+x_\varepsilon )\) converges in \(H^1(\mathbb {R}^3,\mathbb {C}^4)\) to a least energy solution u of
$$\begin{aligned} i\sum \limits _{k=1}^3\alpha _k\partial _k u-a\beta u+V(x_0)u=P(x_0)f(|u|)u+Q(x_0)g(|u|)u. \end{aligned}$$中文翻译：
关于Dirac方程的半经典状态
本文旨在研究具有临界非线性的\（\ mathbb {R} ^ 3 \）中平稳Dirac方程解的存在性和集中性：
$$ \ begin {aligned} i \ varepsilon \ sum \ limits _ {k = 1} ^ 3 \ alpha _k \ partial _k ua \ beta u + V（x）u = P（x）f（| u |）u + Q（x）g（| u |）u，\ end {aligned} $$其中\（\ varepsilon> 0 \）是一个小参数，而\（a> 0 \）是一个常数。我们还示出了半经典的解决方案\（\欧米加_ \ varepsilon \）与最大值点\（X_ \ varepsilon \）浓缩在一组特殊的\（{\ mathcal {H}} _ {P} \），其特征在于V（X）， P（x）和Q（x），以及任意序列\ {x_ \ varepsilon \ rightarrow x_0 \ in {\ mathcal {H}} _ {P}，v_ \ varepsilon（x）：= \ omega _ \ varepsilon（\ varepsilon x + x_ \ varepsilon）\）收敛于\（H ^ 1（\ mathbb {R} ^ 3，\ mathbb {C} ^ 4）\）至最小能量解你的
$$ \ begin {aligned} i \ sum \ limits _ {k = 1} ^ 3 \ alpha _k \ partial _k ua \ beta u + V（x_0）u = P（x_0）f（| u |）u + Q （x_0）g（| u |）u。\ end {aligned} $$
京公网安备 11010802027423号