贺书文,文小波,苑东磊.一类非线性Schrdinger耦合系统的正解[J].重庆工商大学学报(自然科学版),2020,37(4):28-33
HE Shu-wen,WEN Xiao-bo,YUAN Dong-lei.Positive Solutions of a Class of Nonlinear Schrdinger Coupled Systems[J].Journal of Chongqing Technology and Business University(Natural Science Edition),2020,37(4):28-33
一类非线性Schrdinger耦合系统的正解
Positive Solutions of a Class of Nonlinear Schrdinger Coupled Systems
  
DOI:
中文关键词:  Schrdinger耦合系统  变分法  非平凡正解
英文关键词:Schrdinger coupled systems  variational method  non trivial positive solution
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作者单位
贺书文,文小波,苑东磊 1.四川民族学院 理工学院四川 康定 6260012.西南大学 数学与统计学院重庆 400715 
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中文摘要:
      非线性 Schrdinger 方程被广泛应用于数学物理问题中的量子力学、非线性光学等领域,其中非线性 Schrdinger 耦合系统已成为研究热点,对该系统优化和改进非线性项的条件和带周期函数问题是其中比较困难的部分,针对这种定义在无界区域上的耦合问题,提出了一类带多个不同周期函数的非线性 Schrdinger 耦合系统方程;基于变分法和一些分析技巧,将求该类系统的解转化为求对应能量泛函的临界点问题;当该类系统满足适当条件时,可以验证其能量泛函满足山路几何结构,得到一组有界非负的(Ce)c序列,再利用集中紧性原理分两种情形得到其非平凡非负解的存在性;最后由强极大值原理获得该类系统正解的存在性,推广了已有的研究结果。
英文摘要:
      The nonlinear Schrdinger equations are widely used in the fields of quantum mechanics, nonlinear optics, etc. in mathematical physics problems, the nonlinear Schrdinger coupled systems have become a research hotspot. The conditions for the optimization and improvement of the nonlinear terms and the periodic function problems are among more difficult parts, a class of nonlinear Schrdinger coupled system equations with multiple periodic functions is proposed for the coupling problem of this definition on unbounded regions.Based on variational method and some analytical techniques, the solutions of this kind of systems are transformed into the critical points problem of the corresponding energy functional; when the system satisfies the appropriate conditions, it can be verified that the energy functional satisfies the mountain geometry, and a set of bounded non negative (Ce)c sequences is obtained and reused. The compactness principle is used in two cases to obtain the existence of non trivial non negative solutions.Finally, the existence of positive solutions of such systems is obtained by the strong maximum principle,it promotes the existing research results.
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