| 张健,唐先华,张文.一类非线性Maxwell-Dirac系统的驻波解[J].数学年刊A辑,2017,38(1):001~12 |
| 一类非线性Maxwell-Dirac系统的驻波解 |
| Stationary Solutions for a Class of Nonlinear Maxwell-Dirac System |
| Received:May 24, 2016 Revised:October 15, 2016 |
| DOI：10.16205/j.cnki.cama.2017.0001 |
| 中文关键词: Maxwell-Dirac system, Stationary solutions, Strongly indefinite functionals, Variational method |
| 英文关键词:Maxwell-Dirac system, Stationary solutions, Strongly indefinite functionals, Variational method |
| 基金项目:本文受到国家自然科学基金(No.11601145,No.11571370,No.11471137,No.61472136)
和湖南商学院青年教师创新驱动计划(No.16QD008)的资助. |
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| 中文摘要: |
| 对如下非线性Maxwell-Dirac系统
\begin{align*}
\left\{\!\!\!
\begin{array}{ll}
\sum\limits^{3}_{k=1}\alpha_{k}(-\rmi \partial_{k}+K(x)A_{k})u + a\beta u + M(x)u-K(x)A_{0}u =G_{u}(x,u),\-\Delta A_{0}=4\pi K(x)|u|^{2},\-\Delta A_{k}=4\pi K(x)(\alpha_{k}u)\ov{u},\quad k=1,2,3
\end{array}
\right.
\end{align*}
进行了研究, 其中$x\in \mathbb{R}^{3}$. 由于\,Dirac\,算子是上方和下方无界, 相应的能量泛函是强不定的.
假设非线性项满足次临界超二次的增长条件, 运用强不定泛函的广义环绕定理, 证明了系统驻波解的存在性. |
| 英文摘要: |
| This paper is concerned with the following Maxwell-Dirac system
\begin{align*}
\left\{\!\!\!
\begin{array}{ll}
\sum\limits^{3}_{k=1}\alpha_{k}(-\rmi \partial_{k}+K(x)A_{k})u + a\beta u + M(x)u-K(x)A_{0}u =G_{u}(x,u),\\[2mm]
-\Delta A_{0}=4\pi K(x)|u|^{2},\-\Delta A_{k}=4\pi K(x)(\alpha_{k}u)\ov{u},\quad k=1,2,3 \\end{array}
\right.
\end{align*}
for $x\in \mathbb{R}^{3}$. The Dirac operator is unbounded from below and above,
so the associated energy functional is strongly indefinite.\;By applying a
generalized linking theorem for strongly indefinite functionals, the authors establish
the existence of stationary solutions for superquadratic and subcritical nonlinearity. |
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