具有周期外力场Dirac方程的基态解  

Ground state solutions to the Dirac equation with the periodic external force field

作　　者：陈鹏 吴艳[2] 唐先华 Peng Chen;Yan Wu;Xianhua Tang

机构地区：[1]三峡数学研究中心,宜昌443002 [2]三峡大学理学院,宜昌443002 [3]中南大学数学与统计学院,长沙410083

出　　处：《中国科学：数学》2024年第5期747-764,共18页Scientia Sinica：Mathematica

基　　金：国家自然科学基金(批准号:11301297);湖北省自然科学基金(批准号:2021CFB473)资助项目。

摘　　要：本文研究非线性Dirac方程-i∑k=13αk?ku+aβu+M(x)u=g(x,|u|)u基态解的存在性,其中位势函数M(x)是周期的.当非线性项g在无穷远处分别满足超二次与局部超二次增长条件时,利用非Nehari流形方法,在非线性项没有严格单调条件的情形下,证明Nehari-Pankov型基态解的存在性.主要克服了两个困难:(1)相关能量泛函是强不定的,即工作空间分解成的正负子空间的维数都是无穷大,这导致经典的临界点定理不能直接应用;(2)当非线性项不是全局超二次时,验证Cerami序列的环绕结构并证明其有界性.In this paper,we study the ground state solution to the Dirac equation-iZi=1 akOku+au+M(α)u=g(c,lul)u,where the potential function M(α)is periodic when the nonlinear function g satisfies the super-quadratic and local super-quadratic growth conditions at infinity respectively.The existence of Nehari-Pankov-type ground states is proved using the non-Nehari manifold method without strictly monotone conditions.We primarily overcome two major difficulties:(1)the strongly indefinite associated functional,i.e.,the workspace is decomposed into positive and negative subspaces,both with infinite dimensions,which makes the classical critical point theorem unable to be directly applied;(2)verification of the link geometry and the boundedness of Cerami sequences when the nonlinearity is not globally super-quadratic at infinity.

关 键 词：基态解 DIRAC方程 非Nehari流形方法 

分 类 号：O175[理学—数学]