Calculus of Variations and Partial Differential Equations ( IF 2.1 ) Pub Date : 2021-05-10 , DOI: 10.1007/s00526-021-01935-5 Abderrazek Benhassine
This paper is concerned with the following Maxwell–Dirac system
$$\begin{aligned}&-i\sum ^3_{k=1}\alpha _{k}\partial _{k}u + (V(x)+a)\beta u + \omega u-K(x)\phi u =F_u(x,u),\\&\quad -\Delta \phi =4\pi K(x)|u|^2, \end{aligned}$$in \(\mathbb {R}^3\), where V(x) is a potential function and F(x, u) is a nonlinear function modeling various types of interaction and K(t, x) is the varying pointwise charge distribution. Since the effects of the nonlocal term, we use some special techniques to deal with the nonlocal term. Moreover, we prove the existence of infinitely many geometrically distinct solutions for superquadratic as asymptotically quadtratic nonlinearities via variational approach. Some recent results in the literature are generalized and significantly improved. Some examples are also given to illustrate our main theoretical results.
中文翻译：
Maxwell–Dirac系统的驻波解决方案
本文涉及以下麦克斯韦-狄拉克系统
$$ \ begin {aligned}＆-i \ sum ^ 3_ {k = 1} \ alpha _ {k} \ partial _ {k} u +（V（x）+ a）\ beta u + \ omega uK（x ）\ phi u = F_u（x，u），\\＆\ quad-\ Delta \ phi = 4 \ pi K（x）| u | ^ 2，\ end {aligned} $$在\（\ mathbb {R} ^ 3 \）中，其中V（x）是一个势函数，而F（x， u）是一个非线性函数，用于建模各种类型的交互作用，而K（t， x）是变化的逐点电荷分布。由于非本地术语的影响，我们使用一些特殊的技术来处理非本地术语。此外，我们通过变分法证明了超二次渐近二次非线性存在无穷多个几何上不同的解。文献中的一些最新结果得到了概括和显着改善。还给出了一些例子来说明我们的主要理论结果。
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